Coherent Light-Matter Interactions in Monolayer Transition-Metal Dichalcogenides

Coherent Light-Matter Interactions in Monolayer Transition-MetalDichalcogenides

Edbert Jarvis Sie

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Edbert Jarvis Sie

Coherent Light-MatterInteractions in MonolayerTransition-MetalDichalcogenides

Doctoral Thesis accepted by Massachusetts Institute ofTechnology, Cambridge, MA, USA

Edbert Jarvis SieStanford UniversityStanford, CA, USA

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Supervisor’s Foreword

Monolayer transition metal dichalcogenides (TMDs) such as MoS2 and WS2 are

prime examples of atomically thin semiconducting crystals that exhibit remarkable

electronic properties. These materials have generated an enormous amount of

interest in the quantum materials community because, like graphene, they possess

an additional valley degree of freedom (DOF) that has the potential to be used as an

information carrier in next-generation electronics. Unlike graphene, the presence of

broken inversion symmetry in these materials induces bandgaps in each valley that

can be selectively populated with carriers through optical excitation, even though

the two valleys remain to be energetically degenerate locked by time-reversal

symmetry.

Dr. Edbert Jarvis Sie’s thesis made several key contributions to our understand-

ing of light-matter interactions in these materials. By applying coherent ultrafast

optical spectroscopy, Dr. Sie was not only able to study the dynamics of excitations

in these systems but also to optically control their properties.

In particular, he has developed an optical technique to control the valley DOF.

To do this, the degeneracy between the two valleys must first be lifted by breaking

time-reversal symmetry. While applying a magnetic field should in principle

accomplish this task, experiments so far show merely 1–2 meV of valley energy

splitting with fields accessible in common laboratories. Dr. Sie used intense circu-

larly polarized light pulses that are tuned slightly below the bandgap to mimic the

effect of a magnetic field. The light pulse dresses the electronic states such that the

state repulsion between these dressed states leads to a valley-dependent energy shift

(now known as the valley-selective optical Stark effect). He measured the energy

splitting between the two valleys induced by the circularly polarized pump pulse

with a spectrally broadband probe pulse by recording the energy shift of the exciton

absorption peak.

By using this technique on monolayer WS2, he demonstrated that the exciton

energy level in each valley can be selectively tuned because the optical Stark effect

in this material also obeys the valley selection rules with circularly polarized light.

He was able to achieve an energy splitting of 18 meV, which is extremely large,

v

comparable to what would be obtained using a magnetic field of ~100 Tesla.

This finding represents a clear demonstration of breaking the valley degeneracy

in a monolayer TMD, and it offers a novel way of controlling the valley DOF

using off-resonance circularly polarized light for next-generation valleytronic

applications.

After discovering the valley-selective optical Stark effect in TMDs by exciting

with slightly off-resonance light, Dr. Sie decided to study what happens in the case

of large detuning. When light with frequency ω is detuned away from a resonance

ω0, repulsion between the photon-dressed (Floquet) states can lead to a shift of

energy resonance. The dominant effect is the optical Stark shift (/ 1/(ω0�ω)),but there is an additional contribution from the so-called Bloch-Siegert shift

(/ 1/(ω0 +ω)). Although it is common in atoms and molecules, the observation

of the Bloch-Siegert shift in solids has so far been limited only to artificial atoms

since the shifts were small (<1 μeV) and inseparable from the optical Stark shift.

Dr. Sie observed an exceptionally large Bloch-Siegert shift (~10 meV) in mono-

layer WS2 under infrared optical driving. Moreover, he was able to disentangle the

Bloch-Siegert shift entirely from the optical Stark shift, because the two effects are

found to obey opposite selection rules at different valleys. By controlling the light

helicity, he was able to confine the Bloch-Siegert shift to occur only at one valley

and the optical Stark shift at the other valley. This discovery is quite important since

such a valley-exclusive Bloch-Siegert shift allows for enhanced control over the

valleytronic properties in two-dimensional materials.

Dr. Sie also presents a number of other key results especially on many-body

interactions between excitons in TMDs. For example, interactions between two

excitons can result in the formation of bound quasiparticles, known as biexcitons.

He used a clever optical trick to observe intervalley biexcitons that comprise two

excitons from different valleys. These novel quasiparticles have no analogue in

conventional semiconductors. Dr. Sie also managed to observe a new type of

optical Stark effect in monolayer WS2, one that is mediated by such intervalley

biexcitons under a blue-detuned driving with circularly polarized light. At higher

excitation densities, many-body interactions between excitons are expected to lead

to several other phenomena that have been quite challenging to observe in conven-

tional semiconductors. Dr. Sie found in monolayer WS2 that the exciton resonance

energy exhibits a pronounced redshift followed by an anomalous blueshift at

increasing exciton density. This observation reveals an attraction-repulsion cross-

over of interactions between excitons, which mimics the well-known Lennard-

Jones interactions between atoms.

In summary, this thesis provides many key insights into light-matter interactions

in monolayer TMDs. Coherent ultrafast optical spectroscopy was successfully used

to both probe and control excitations in these atomically thin semiconductors.

These results may pave the way for enhanced optical control over the valleytronic

properties in two-dimensional materials.

Massachusetts Institute of Technology

Cambridge, MA, USA

Nuh Gedik

vi Supervisor’s Foreword

Preface

Semiconductors that are thinned down to a few atomic layers can exhibit novel

properties beyond those encountered in bulk forms. Transition-metal

dichalcogenides (TMDs) such as MoS2, WS2, and WSe2 are prime examples of

such semiconductors. They appear in layered structure that can be reduced to a

stable single layer where remarkable electronic properties can emerge. Monolayer

TMDs have a pair of electronic valleys which have been proposed as a new way to

carry information in next-generation devices, called valleytronics. However, these

valleys are normally locked in the same energy level, which limits their potential

use for applications.

This dissertation presents the optical methods to split their energy levels by

means of coherent light-matter interactions. Experiments were performed in a

pump-probe technique using a transient absorption spectroscopy on MoS2 and

WS2 and a newly developed XUV light source for time- and angle-resolved

photoemission spectroscopy (TR-ARPES) on WSe2 and WTe2. Hybridizing the

electronic valleys with light allows us to optically tune their energy levels in a

controllable valley-selective manner. In particular, by using off-resonance circu-

larly polarized light at small detuning, we can tune the energy level of one valley

through the optical Stark effect. At larger detuning, we observe a separate contri-

bution from the so-called Bloch-Siegert effect, a delicate phenomenon that has

eluded direct observation in solids. The two effects obey opposite selection rules,

which enables us to separate the two effects at two different valleys.

Monolayer TMDs also possess a strong Coulomb interaction that enhances

many-body interactions between excitons, both bonding and nonbonding interac-

tions. In the former, bound excitonic quasiparticles such as biexcitons play a unique

role in coherent light-matter interactions where they couple the two valleys to

induce opposite energy shifts. The latter are found to exhibit energy shifts that

vii

effectively mimic the Lennard-Jones interactions between atoms. Through these

works, we demonstrate new methods to optically tune the energy levels of elec-

tronic valleys in monolayer TMDs.

Stanford, CA, USA Edbert Jarvis Sie

viii Preface

Parts of this Thesis Have Been Published in the Following Journal Articles

1. E.J. Sie, C.H. Lui, Y.H. Lee, J. Kong, N. Gedik, Large, valley-exclusive Bloch-

Siegert shift in monolayer WS2, Science 355, 1066–1069 (2017) (pdf)2. E.J. Sie, J.W. McIver, Y.H. Lee, L. Fu, J. Kong, N. Gedik, Valley-selective

optical Stark effect in monolayer WS2, Nat. Mater. 14, 290–294 (2015) (pdf)

(Cover Story)

3. E.J. Sie, A. Steinhoff, C. Gies, C.H. Lui, Q. Ma, M. R€osner, G. Sch€onhoff,F. Jahnke, T.O. Wehling, Y.H. Lee, J. Kong, P. Jarillo-Herrero, N. Gedik,

Observation of exciton redshift-blueshift crossover in monolayer WS2, Nano

Lett. 17, 4210–4216 (2017) (pdf)

4. E.J. Sie, C.H. Lui, Y.H. Lee, J. Kong, N. Gedik, Observation of intervalley

biexcitonic optical Stark effect in monolayer WS2, Nano Lett. 16, 7421–7426

(2016) (pdf)

5. E.J. Sie, A.J. Frenzel, Y.H. Lee, J. Kong, N. Gedik, Intervalley biexcitons and

many-body effects in monolayer MoS2, Phys. Rev. B 92, 125,417 (2015) (pdf)

6. E.J. Sie, J.W. McIver, Y.H. Lee, L. Fu, J. Kong, N. Gedik, Optical Stark effect in

2D semiconductors, Proc. SPIE 9835 (Invited Paper), Ultrafast Bandgap Pho-tonics, 983,518 (2016) (pdf)

ix

http://science.sciencemag.org/content/355/6329/1066

http://www.nature.com/nmat/journal/v14/n3/full/nmat4156.html

http://www.nature.com/nmat/journal/v14/n3/covers/index.html

http://pubs.acs.org/doi/abs/10.1021/acs.nanolett.7b01034

http://pubs.acs.org/doi/abs/10.1021/acs.nanolett.6b02998

http://journals.aps.org/prb/abstract/10.1103/PhysRevB.92.125417

http://proceedings.spiedigitallibrary.org/proceeding.aspx?articleid=2523979

Acknowledgments

First and foremost, I would like to thank my advisor, Prof. Nuh Gedik, for his

mentorship and support. He has been a constant source of inspiration since I first

started my graduate studies. In fact, it was even before that. I remember our first

meeting at a conference in Boston, where he presented a real-time image of

crystalline lattice motion, which was an awe-inspiring moment for me. He has

been a tremendous support since day one, when I was given the opportunity to

design and build the extreme ultraviolet (XUV) beamline for ARPES in the lab as

well as the freedom to build the white light setup next to it. His vigor for innova-

tions in time-resolved experiments is always inspiring. Throughout my graduate

studies, he has been a true role model for how to be a good scientist and to search for

answers to questions like “So what?” in a perfectly well-stated manner. I also

learned that “no problem is too small, and no problem is too big.” He gave me

and other group members many opportunities to attend international conferences

with the intention that we could learn how to communicate our findings with new

people and meet fellow scientists in the field. Despite his busy schedule, he always

makes time to discuss with group members regularly through weekly group meet-

ings, bi-weekly subgroup meetings, and individual meetings. Apart from discussing

research, he has been a constant source of encouragement and guidance, chatting

with us about personal matters and our careers. Sometimes, I was left wondering

how one person can do so much in such little time, and can become as quick and

sharp minded as a laser while remaining gentle at all times. I sincerely thank Nuh

for every aspect of my graduate education.

I am very grateful to other members of my thesis committee, Prof. Liang Fu and

Prof. Pablo Jarillo-Herrero. Liang has been my theory advisor; many fundamental

explanations for my experiments are inspired by him. One of the many insightful

wisdoms he imparted to me is that the optical Stark effect can be extended to create

a topologically nontrivial phase of matter in monolayer TMDs. Discussing research

with him has always been exciting and stimulating, not because we know anything

more about the theory, but because he knows very well the reality about what the

experiments can do. Much of the theory I understand was patiently explained to me

xi

after I visited his office several times. He showed me how to use simple diagrams

and mathematical expressions that capture the essence of physics. Now I remember

to use Cartesian coordinates every time I face spherical harmonics problems.

Pablo has been my academic advisor at MIT, where he regularly checked on my

academic progress throughout my graduate studies. His office is like a holy and

respected temple where I can freely discuss my classes, research progress, and

academic career and seek his guidance. I always remember his words that it is better

to complete tasks one at a time rather than attempting to complete everything at

once. I sincerely thank Pablo for letting me have access to his labs to borrow optics

or to perform sample characterization with his group. Every time I invited his group

members out for lunch or dinner, I always feel grateful to be among such highly

skilled people capable of creating cool devices.

My learning and working experience in Gediklab has been very enjoyable

because of the highly spirited colleagues. I am very grateful to Yihua Wang (now

professor at Fudan University), who taught me “Gediklab 101” when I first joined

the group in August 2011. He constructed the ARPES system in the lab and taught

me how it works and how to use it. I learned how to use SolidWorks from him in

order to accurately design and assemble many ultrahigh vacuum (UHV) appara-

tuses for the new extreme ultraviolet (XUV) beamline in the lab. Since then, it has

become my favorite software to draw three-dimensional figures; some of them

appear in the last chapter of this dissert. David Hsieh (now professor at Caltech) has

always been inspiring, and he introduced me to many interesting material systems.

Timm Rohwer has been a great colleague that I had a real pleasure to work together

with and a good friend to have. He joined the group during my third year, and since

then, we have been working together to set up the XUV beamline for ARPES. He

has an infinite reservoir of energy and motivation, which constantly inspires

me. There is no lab equipment that he cannot fix. Zhanybek Alpichshev has been

a constant source of joy and inspiration in the office. I am very fortunate to share an

office with him. I learn something new almost every time I have a conversation with

him. There is no question that he cannot answer. Joshua Lui (now professor at UC

Riverside) taught me how to clearly write stories worth telling. He gave me very

helpful advice during my transition from a graduate student into a more devoted

professional in a later stage of my career.

I also enjoyed my discussions with former group members James McIver, Alex

Frenzel, Darius Torchinsky, Daniel Pilon, Fahad Mahmood, Byron Freelon, Inna

Vishik, Hengyun Zhou, Mahmoud Ghulman, and Steven Drapcho, as well as with

current group members Ozge Ozel, Changmin Lee, Emre Ergecen, Alfred Zong,

Anshul Kogar, Carina Belvin, Suyang Xu, Nikesh Koirala, Mehmet Yilmaz, Guy

Marcus, and Edoardo Baldini. They are very talented individuals and enjoyable to

work with in the lab. There were many good moments that we shared together,

which enriched my experience as a graduate student at MIT.

Outside our lab, I am very thankful to Prof. Yi-Hsien Lee (now professor at

NTHU, Taiwan) and Prof. Jing Kong for their expertise and generous support to

synthesize the high-quality monolayer TMDs that I used in all of my experiments.

My first encounter with Yi-Hsien in the corridor of MIT Building 13 resulted in a

xii Acknowledgments

long-lasting friendship and professional collaboration, which I hope will continue

in future. I would like to thank Wenjing Fang, who happens to be the source of all

interesting information surrounding our community. I remember her telling me

about the new thin semiconducting materials that could be synthesized in Jing’sgroup with the help from Yi-Hsien. This started my pursuit in studying the

monolayer TMDs. I am very grateful to my wonderful friend and colleague

Qiong Ma from Pablo’s group for being my study buddy over the years. She

shows great, professional enthusiasm at work (and outside), which inspires me to

work harder and be more productive. Every time I have questions regarding

graphene or low-dimensional devices, she has always been the first person I

consulted with, and she usually gave more useful tips beyond my questions. She

is like a living dictionary made of graphene, a strong one that can even hold an

elephant! Everybody knows this is no joke. I also thank Yaqing Bie and Yuan Cao

for being my dinner buddies whenever I have a craving for good food (like every

time). Yaqing, the “Queen of Optics” in Pablo’s group, has always been helpful

every time I wanted to discuss about optics in low-dimensional devices. I also thank

the rest of the people in MIT Building 13, Yafang Yang, Linda Ye, Shengxi Huang,

Yuxuan Lin, Lin Zhou, and Xi Ling (now professor at Boston University), who

have been amazing companions during my graduate studies, as well as in exploring

oriental cuisine. I thank Raga Markely for being my lunch buddy, with whom I can

also share my thoughts about everyday life and career. I also thank Yichen Shen,

Purnawirman, Takuma Inoue, Long Wu, Ken Xuan Wei, Chong-U Lim, and Sang

Woo Jun who joined the PhD program at MIT the same year as me and for our

friendship over the years. I am very grateful to Monica Wolf, who has done a lot

more than being our admin secretary. She took care of everything: travel plans,

reimbursements, lab equipment purchases, employment transition, and the big

celebration for my thesis defense (with the help from Qiong and Ozge).

I would like to thank Christopher Gies and Alexander Steinhoff in the University

of Bremen, Germany, for their immense contributions in the theoretical study of

highly excited monolayer TMDs, presented in Chap. 7. It was a wonderful experi-

ence to collaborate with Chris and Alex, from whom I learned a great deal about the

role of plasma screening in monolayer TMDs.

Throughout the years as a graduate student, there were also other researchers

I often encountered at conferences: Jonghwan Kim, Long Ju, Xiaoxiao Zhang,

Yilei Li, Ziliang Ye, Jan Buss, Akshay Singh, and Shiang Fang. I am very grateful

to know them; they make the conference visits more interesting and fruitful,

because we work in similar fields and we can learn from each other. I am sure I

will see them more often in the future, and I look forward to it.

My very special gratitude goes to my parents for their unconditional love,

support, and wisdom. My mother taught me how to speak, to stand, and to love

one another. My father fixed what my mother had taught me.

Edbert Jarvis SieMassachusetts Institute of Technology

Cambridge, MA, USA

Acknowledgments xiii

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Monolayer Transition Metal Dichalcogenides (TMDs) . . . . . . . . . 2

1.1.1 Electronic Band Structure . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Optical Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.3 Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Time-Resolved Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Coherent Light-Matter Interactions . . . . . . . . . . . . . . . . . . 9

1.2.2 Quasi-equilibrium Dynamics . . . . . . . . . . . . . . . . . . . . . . 9

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Time-Resolved Absorption Spectroscopy . . . . . . . . . . . . . . . . . . . . . 13

2.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.2 Laser Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.3 Optical Parametric Amplifier . . . . . . . . . . . . . . . . . . . . . . 16

2.1.4 White Light Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Kramers-Kronig Analysis . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.2 Maxwell’s Equations for Monolayer Sample . . . . . . . . . . . 22

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Intervalley Biexcitons in Monolayer MoS2 . . . . . . . . . . . . . . . . . . . . 27

3.1 Intervalley Biexcitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 Experimental Results and Discussions . . . . . . . . . . . . . . . . . . . . . 30

3.3.1 Intravalley and Intervalley Scattering . . . . . . . . . . . . . . . . 31

3.3.2 Signature of Intervalley Biexcitons . . . . . . . . . . . . . . . . . . 32

3.4 Time-Resolved Cooling Process . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

xv

4 Valley-Selective Optical Stark Effect in Monolayer WS2 . . . . . . . . . 37

4.1 Optical Stark Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1.1 Semiclassical Description . . . . . . . . . . . . . . . . . . . . . . . . 40

4.1.2 Quantum-Mechanical Description . . . . . . . . . . . . . . . . . . 42


4.3 Observation of the Optical Stark Effect . . . . . . . . . . . . . . . . . . . . 44

4.4 Valley Selectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.5 Fluence and Detuning Dependences . . . . . . . . . . . . . . . . . . . . . . 46

4.6 Proposal: Valley-Specific Floquet Topological Phase

in TMDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.7 Supplementary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.7.1 Time-Resolved Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.7.2 Polarization-Resolved Spectra . . . . . . . . . . . . . . . . . . . . . 51

4.7.3 Obtaining Energy Shift . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.7.4 Comparison from Semiclassical Theory . . . . . . . . . . . . . . 55

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5 Intervalley Biexcitonic Optical Stark Effect

in Monolayer WS2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.1 Blue-Detuned Optical Stark Effect . . . . . . . . . . . . . . . . . . . . . . . 60


5.3 Experimental Results and Data Analysis . . . . . . . . . . . . . . . . . . . 61

5.4 Intervalley Biexcitonic Optical Stark Effect . . . . . . . . . . . . . . . . . 64

5.4.1 Four-Level Jaynes-Cummings Model . . . . . . . . . . . . . . . . 66

5.5 Perspective: Zeeman-Type Optical Stark Effect . . . . . . . . . . . . . . 68

5.6 Supplementary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.6.1 Coherent and Incoherent Optical Signals . . . . . . . . . . . . . 68

5.6.2 Time-Trace Fitting Decomposition Analysis . . . . . . . . . . . 70

5.6.3 Possible Effects Under Red-Detuned Pumping . . . . . . . . . 72

5.6.4 Fitting Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6 Large, Valley-Exclusive Bloch-Siegert Shift

in Monolayer WS2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.1 Bloch-Siegert Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.1.1 Semiclassical Description . . . . . . . . . . . . . . . . . . . . . . . . 79

6.1.2 Quantum-Mechanical Description . . . . . . . . . . . . . . . . . . 82


6.3 Observation of the Bloch-Siegert Shift . . . . . . . . . . . . . . . . . . . . 85

6.4 Fluence and Detuning Dependences . . . . . . . . . . . . . . . . . . . . . . 87

6.5 Valley-Exclusive Optical Stark Shift and Bloch-Siegert

Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

xvi Contents

7 Lennard-Jones-Like Potential of 2D Excitons

in Monolayer WS2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.1 Many-Body Interactions in 2D TMDs . . . . . . . . . . . . . . . . . . . . . 94


7.3 Optical Signature of Many-Body Effects . . . . . . . . . . . . . . . . . . . 95

7.3.1 Exciton Redshift-Blueshift Crossover . . . . . . . . . . . . . . . . 96

7.3.2 At Low Density: Plasma Contribution . . . . . . . . . . . . . . . 98

7.3.3 At High Density: Exciton Contribution . . . . . . . . . . . . . . . 99

7.4 Lennard-Jones-Like Potential as an Effective Model . . . . . . . . . . 99

7.5 Chronological Signature of Interactions

in Time-Resolved Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.7 Supplementary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.7.1 Microscopic Many-Body Computation . . . . . . . . . . . . . . . 104

7.7.2 Exciton-Exciton Annihilation Effect . . . . . . . . . . . . . . . . . 107

7.7.3 Heat Capacity and Estimated Temperature . . . . . . . . . . . . 108

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

8 XUV-Based Time-Resolved ARPES . . . . . . . . . . . . . . . . . . . . . . . . . 115

8.1 Building a High-Resolution XUV Light Source

for TR-ARPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8.1.2 XUV Light Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

8.1.3 XUV Monochromator . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8.1.4 XUV Diagnostic Chamber . . . . . . . . . . . . . . . . . . . . . . . . 122

8.2 Measuring TMDs Using 30 eV XUV TR-ARPES . . . . . . . . . . . . 123

8.2.1 WSe2 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

8.2.2 WTe2 Semimetal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Contents xvii

Chapter 1

Introduction

Two-dimensional materials can exhibit novel properties beyond those normally

encountered in the bulk compounds. Most evidently is the change in their electronic

density of states and their increasing susceptibility to a Peierls transition. More

interestingly, a number of fascinating phenomena emerge from two-dimensional

materials. This includes quantum well system with quantized energy levels,

two-dimensional electron gas (2DEG) that can exhibit the integer and fractional

quantum Hall effect, layered materials that demonstrate high-temperature super-

conductors, as well as topological insulators that feature nontrivial metallic states at

the surface while remain insulating on the inside. Some of these material systems

can appear as compounds from multiple different elements with complex lattice

structures while maintaining their quasi-two-dimensionality.

Graphene, a single layer of carbon atoms, is a prime example of such 2D

materials that can exhibit some of the above remarkable properties despite its

very simple structure. The electronic structure of graphene is described by a

massless Dirac fermion at around the Fermi level, with the valence and conduction

© Springer International Publishing AG 2018

E.J. Sie, Coherent Light-Matter Interactions in Monolayer Transition-MetalDichalcogenides, Springer Theses, https://doi.org/10.1007/978-3-319-69554-9_1

1

bands touching at a single point [1]. This allows graphene to exhibit extremely high

electron mobility, which is a promising property in developing nanoscale electronic

applications. However, the gapless electronic structure in graphene makes it chal-

lenging to use this material for switchable electronics such as transistors. Various

approaches can be used to open a gap in graphene [2], for example, by placing

graphene above a particular substrate that breaks graphene’s sublattice symmetry.

However, the resulting gap induced this way is reported to be around 100 meV [3],

which is still rather small for practical applications.

Transition metal dichalcogenides (TMDs) comprise a family of II–VI semi-

conductors such as MoS2, WS2, MoSe2, and WSe2. These materials are layered in

structure, from which we can isolate a monolayer of TMDs through mechanical

exfoliation or chemical vapor deposition (CVD) growth method [4, 5]. Monolayer

TMDs have a lattice structure similar to graphene, but, unlike graphene, these

materials possess a semiconducting gap between 1 and 2 eV. The large gap in

this class of materials offers promising electronic applications based on

two-dimensional materials. Recently, a microprocessor based on a 2D semiconduc-

tor (MoS2) has been reported [6]. The device consists of 115 transistors comprising

all basic building blocks that are common to most microprocessors. While this class

of materials could be an important element for the development of future 2D

devices, they are found to have novel electronic and optical properties as we discuss

below.

1.1 Monolayer Transition Metal Dichalcogenides (TMDs)

The properties of monolayer TMDs were first studied through pioneering optical

measurements on monolayer MoS2 by Mak et al. [7] and Splendiani et al.

[8]. Monolayer MoS2 is a one-unit-cell-thick semiconductor with a graphene-like

hexagonal lattice of Mo and S atoms where the S-Mo-S layers are stacked in a

trigonal prismatic arrangement (Fig. 1.1a, b). It is a direct bandgap semiconductor

with an optical energy gap of 1.96 eV (10 K) at K and K0 valleys in the Brillouin

zone and a valence band splitting of 160 meV due to strong spin-orbit coupling

(SOC) (Fig. 1.2). Other monolayer TMDs have similar properties, but the magni-

tudes differ slightly, for example, monolayer WS2 has an optical gap of 2.1 eV at

300 K and a valence band splitting of 400 meV. Monolayer TMDs are promising

material systems for future low-energy electronics because it has a unique spin-

valley coupling and a strong Coulomb interaction.

1.1.1 Electronic Band Structure

In this section, we describe the electronic band structure of monolayer MoS2,

following the theoretical studies by Xiao et al. [9] and Liu et al. [10, 11], but the

2 1 Introduction

general description should also apply to WS2, WSe2, and MoSe2 monolayers. The

band structure of monolayer MoS2 around the gap is well described by considering

the Mo-d orbitals and S-p orbitals. This material has a unit cell that forms a trigonal

prismatic coordination around the Mo atom that splits its d orbitals into three

groups: A1 dz2ð Þ,E dxy; dx2�y2� �

, and E0(dxz, dyz). The mirror symmetry along the z

direction allows a hybridization between A1 and E orbitals, which constitutes the

gap opening at the K and K0 valleys in the Brillouin zone (Fig. 1.3). In the vicinity

of these symmetry points, the wave function of the bands is described by the

following symmetry-adapted basis functions:

ϕcj i ¼ dz2j i, ϕvj i ¼ 1ffiffiffi2

p dx2�y2�� i dxy

�� ð1:1Þ

where |ϕci is the basis for conduction band and |ϕvi for valence band and the� sign

denotes the K or K0 valley index. The two-band Hamiltonian of monolayer MoS2

Fig. 1.1 (a) Unit cell of monolayer 2H-MoS2 in a trigonal prismatic arrangement. This structure

lacks an inversion center. (b) Top view of monolayer 2H-MoS2

Fig. 1.2 Hexagonal Brillouin zone of monolayer MoS2. The band edges from conduction bands

(CB) and valence bands (VB) are located at the K and K0 points

1.1 Monolayer Transition Metal Dichalcogenides (TMDs) 3

mimics that of graphene because both systems have similar symmetry properties,

except for the gap opening due to the differing on-site potentials between the Mo

and S sublattices and the strong spin-orbit coupling (SOC) originated from the

Mo-d orbitals:

H ¼ at��kxbσx þ kybσy

�þ Δ2bσ z � λ

bσ z � 1

2sz ð1:2Þ

where a is the lattice constant, t the hopping integral, k the wave vector, bσ the Pauli

matrices for the two basis functions,Δ the energy gap, 2λ the spin splitting at the topof the valence band caused by the SOC interactionH

0 ¼ λL � S (ml¼ � 2,ms¼ � 1/2),

and sz the Pauli matrices for the electron’s spin.There are three important features that distinguish monolayer MoS2 from

graphene. First, the resulting electronic structure is described by a massive Dirac

fermion due to the finite gap in the second term. This allows an interband electronic

transition at the K and K0 valleys that can be induced through an optical excitation

in the visible regime. Second, the two valleys possess finite and opposite Berry

curvatures owing to the lack of crystalline inversion symmetry. This gives rise to

distinct optical selection rules, which we will discuss further in the next section.

Third, the SOC contribution in the third term leads to a large spin splitting at the

valence band top (160 meV). Meanwhile, the spin splitting at the conduction band

bottom is negligible in the first-order approximation because it is mostly composed

Fig. 1.3 Atomic orbital projection of band structures for monolayer MoS2 from first-principle

calculations, without spin-orbit coupling. Fermi energy is set to zero. Symbol size is proportional

to its contribution to corresponding state. (a) Contributions from Mo-d orbital. (b) Total contri-

butions from p orbitals, dominated by S atoms. (c) Total s orbitals (Figure is obtained from Ref.

[10])

4 1 Introduction

by dz2 orbital (ml ¼ 0). Its magnitude can be obtained by using a second-order

perturbation from the Mo-d orbitals and a first-order perturbation from the minor

S-p orbitals (ml¼�1) [12]. Although the conduction band splitting is much smaller

in magnitude (3 meV in MoS2), it plays an important role in the formation of stable

dark excitons in monolayers WS2 and WSe2 [13]; the splitting is larger

(20–40 meV) and inverted.

1.1.2 Optical Selection Rules

In monolayer TMDs, the two valleys couple differently with left- and right-

circularly polarized light [9, 14–16]. At K valley, optical transition from the valence

band (VB) top to the conduction band (CB) bottom is induced exclusively by left-

circularly polarized light. At K0 valley, the optical transition is induced exclusivelyby right-circularly polarized light. Such valley selection rules arise from the

different quantum numbers m associated to the bands at the two valleys

(Fig. 1.4). At K valley, the valence bands are assigned m ¼ �1/2 and +1/2 for the

spin-down and spin-up, respectively, while the conduction bands are assigned

m ¼ �3/2 and �1/2 for the spin-down and spin-up, respectively. Interband transi-

tions from VB to CB can occur through absorption of left-circularly polarized light

(σ�) that carries an angular momentum Δm¼ � 1, where we consider electric

dipole approximation that leaves the spin unchanged. At K0 valley, the quantum

numbers have the opposite signs with respect to the K valley because of time-

reversal symmetry. Hence, interband transition at K0 valley can occur through

absorption of right-circularly polarized light (σ+) with Δm¼ + 1.

The above description can explain the origin of valley selection rules because the

quantum numbers are given. However, obtaining these quantum numbers is not as

straightforward as the addition of orbital and spin angular momenta in atoms.

Isolated atoms in free space have a full spherical symmetry that allows electron

orbitals to acquire magnetic quantum numbers m¼ 0, � 1, � 2, � 3, . . .. with no

limit. Meanwhile, the crystal lattice of monolayer TMDs has a space groupD13h that

imposes a trigonal symmetry C3 at the K symmetry points, so the linear combina-

tion of atomic orbitals (LCAO) throughout the lattice must conform this symmetry.

This limits the effective quantum numbers m¼ 0, � 1, � 2 (modulo 3) that can be

present in the system.

In order to obtain the quantum numbers of the conduction and valence bands, we

must find the eigenvalue γα (α ¼ c,v) of the C3 rotation on the Bloch functions

C3ψα¼ γαψα. This should include both the constructing atomic orbitals and the

plane wave component of the Bloch functions, expressed as

ψα rð Þ ¼ 1ffiffiffiffiN

pXR

e�iK� rþδð Þdα r� R� δð Þ ð1:3Þ


where R is the lattice vector, δ is the position of Mo atom in the unit cell, dc ¼ dz2 ,and dv ¼ dx2�y2 � idxy

� �=

ffiffiffi2

p. There are two contributions to the eigenvalue, first

the C3 rotation about the center of atomic orbital C3dc¼ dc and C3dv¼ e�i2π/3dv andsecond the change of lattice phase factor e�iK � r at each lattice site. The eigenvalue

γα depends on the position of the rotation center ρ, which can be Mo atom, S atom,

or the h position (center of hexagon), as shown in Fig. 1.5 (inset). By using

symmetry analysis of C3 on the above Bloch functions, we can obtain the

eigenvalue:

γ ρα ¼ e�iK� C�13 δ�δð Þγdα ð1:4Þ

where δ is the Mo atom position with respect to the rotation center ρ and γdα is theeigenvalue about the center of atomic orbitals [11]. In short, the orbital quantum

number ml is obtained from γ¼ exp(i2πml/3). The simplest example is given at ρ¼Mo(δ¼ 0) where the lattice phase contribution is zero and hence γMo

c ¼ γdc ¼ 1 and

γMov ¼ γdv ¼ e�i2π=3. The conduction bands are assigned ml¼ 0 at K and K0 valleys,while the valence bands are assigned ml ¼ �1 at K valley and +1 at K0 valley. Byadding the spin contributions ms, we can obtain the total quantum numbers mj of

these bands (Fig. 1.5).

We emphasize that the quantum numbers associated with conduction and

valence bands depend on the choice of rotation center, but the upward optical

transition between the spin-conserving bands remains the same, i.e., Δm¼ � 1 at

K valley and +1 at K0 valley. This is a consequence of the lattice symmetry in

monolayer TMDs that also gives rise to the non-zero and opposite Berry curvatures

at the K and K0 valleys. Berry curvature can be viewed as the self-rotating motion of

the electron wave packet or the orbital magnetic moment, and the opposite Berry

curvatures between the K and K0 valleys give rise to the opposite selection rules at

the two valleys.

Fig. 1.4 Optical selection

rules in monolayer MoS2

6 1 Introduction

1.1.3 Excitons

Upon photoexcitation, an electron will occupy the conduction band and leave a hole

in the valence band. They have opposite charges and can attract each other through

Coulomb interaction. This allows a formation of bound quasiparticle called an

exciton. Monolayer TMDs are atomically thin semiconductors; so electronic exci-

tation is confined within 2D structure similar to the situation in quantum wells.

There are two important consequences that differentiate the excitons in this system

and in the bulk. First, there is a strong spatial overlap between the electron and hole

orbitals in the z direction due to quantum confinement. Second, the electric field

between the electron and hole pair can penetrate outside the material where

screening is absent (Fig. 1.6a–c). These two features enhance the Coulomb attrac-

tion significantly and result in the large excitonic binding energy of 320 meV in

monolayer TMD WS2 as compared to 50 meV in bulk WS2 [17]. The binding

energies are determined through comparing the series of exciton energy levels

obtained experimentally and theoretically. Note that as the electron-hole separation

Fig. 1.5 The C3 rotation eigenvalues γc,v [10] and quantum numbers of bands in monolayer MoS2,

specified for different rotation centers ρ¼Mo, S, and h positions (see inset). The quantum numbers

inside the middle row are shown only for the band edges, i.e., spin-down at K valley and spin-up at

K0 valley.


increases, a larger amount of electric field between them penetrates outside the

material and the overall screening gets suppressed. This leads to a pronounced

deviation from the usual hydrogenic series of exciton energy levels.

The optical selection rules in the previous section are discussed in terms of

one-particle picture where Coulomb interaction between the electron and hole pair

is ignored. Note that the selection rules result from the symmetry properties in

monolayer TMDs. Since Coulomb interaction appears as 1/r with a full spherical

symmetry, the symmetry properties in monolayer TMDs are unaffected. Thus,

novel selection rules at the two valleys are maintained, and excitons can be created

in a valley-selective manner using circularly polarized light.

1.2 Time-Resolved Spectroscopy

Monolayer TMDs have a pair of electronic valleys (K and K0) whose index can be

used as a new degree of freedom for next-generation valleytronics. Owing to the

optical selection rules, we can use left-circularly polarized light to promote charge

carriers at K valley to carry information with valley index of �1. However, since

the two valleys are energetically degenerate, these carriers can scatter from K valley

to K0 valley and carry the incorrect valley index of +1 (Fig. 1.7). Such intervalley

scattering introduces a valley index mixture throughout propagation until the

population between the two valleys reaches equilibrium. In order to use the valley

index for practical valleytronic devices, it is desirable to suppress the intervalley

scattering. There are two methods to follow up on this line: (1) split the energy

levels between the two valleys, and (2) identify the dominant intervalley scattering

mechanisms and suppress its scattering rate. In our approach, we use light-matter

interactions in a time-resolved manner to achieve this.

Fig. 1.6 Interactions between electron and hole that form an exciton are mediated by Coulomb

attraction. (a) In bulk, the electric field is screened by the surrounding material. (b) In monolayer,

large fraction of the electric field penetrates outside the material into the vacuum where screening

is absent or to an appropriate substrate where screening can be varied. (c) The suppressed

screening in monolayer leads to a larger exciton binding energy with respect to the bandgap

(Figures are adapted and redrawn from Ref. [17])

8 1 Introduction

1.2.1 Coherent Light-Matter Interactions

Hybridizing light with electrons in materials can result in newly created photon-

dressed states or Floquet states with energy levels that are modified from their

original states. An effective hybridization can be achieved in materials that are

subject to a coherent light field so there exists a phase correlation between the

induced polarization and the incident light. Hence, the resulting hybrid quasiparti-

cle, when written using Bogoliubov transformation, can maintain well-defined

amplitude and energy. This is in a similar situation with the forced harmonic

oscillator, where the interaction energy is transferred periodically between light

and electrons in materials without energy loss. In practice, energy loss can occur

through incoherent interactions. This could be mediated by dissipative scattering

channels upon absorption of resonance light.

In order to suppress the absorption process, we use coherent light source (laser)

that is tuned off-resonance with the material system. Coherent interactions with

off-resonance light can be used to shift the energy levels of materials. As we will

discuss in later chapters, the energy shift (often called the “light shift”) can occur

through two mechanisms, the optical Stark shift and the Bloch-Siegert shift. These

processes can be used to split the energy level between the two valleys using

circularly polarized light that breaks time-reversal symmetry. This optical tech-

nique can offer a much larger energy splitting as compared to that of magnetic field.

1.2.2 Quasi-equilibrium Dynamics

In monolayer TMDs, excitons can be created at a specific valley and switched to the

other valley through intervalley scattering. The lifetime of this scattering process

can be measured using pump-probe spectroscopy, which can occur in

subpicosecond timescale. In this experiment, a left-circularly polarized pump

pulse is used to create excitons at K valley and no excitation at K0 valley. Aftersome time delay, exciton populations at K and K0 valleys are monitored using left-

and right-circularly polarized probe pulse. The intervalley scattering process can be

Fig. 1.7 Intervalley

scattering of charge carriers

from K valley to K0 valley

1.2 Time-Resolved Spectroscopy 9

measured through the increasing bleaching signal at K0 valley in time-resolved

absorption spectroscopy or through the decreasing polarization rotation of the probe

pulse in time-resolved Faraday or Kerr spectroscopy. The obtained scattering

lifetime can be scrutinized through its dependencies in temperature, charge doping,

spin-orbit coupling strength, exciton density, and substrates. As such, the dominant

intervalley scattering mechanism can be identified, and the scattering rate can be

suppressed.

Mutual interactions between excitons can also be probed using time-resolved

spectroscopy. In later chapters, we will discuss the bound state of two excitons,

called biexciton, which can be induced using two-step excitation. Two excitons that

are created at two different valleys can form an intervalley biexciton, which possessa unique combination of orbital, spin, and valley degree of freedoms, with no

analogue in atomic systems. Nonbonding interactions between two separate exci-

tons can also occur through atom-like van der Waals interactions. Unlike atoms,

however, excitons usually have much smaller binding energies and can break into

free electron-hole pairs or plasma. This introduces complexities to nonbonding

interactions between excitons in conventional semiconductors. Monolayer TMDs,

on the other hand, provide a special platform for exploring quasiparticle interac-

tions because of their unique two-dimensionality and strong Coulomb interactions,

which render these quasiparticles stable at room temperature. This should allow

more in-depth studies of such interactions at higher excitation density regime that

was previously unexplored in conventional semiconductors.

References

1. A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, The electronic

properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009)

2. S. Ryu, C. Mudry, C.Y. Hou, C. Chamon, Masses in graphenelike two-dimensional electronic

systems: topological defects in order parameters and their fractional exchange statistics. Phys.

Rev. B 80, 205319 (2009)

3. E. Wang et al., Gaps induced by inversion symmetry breaking and second-generation Dirac

cones in graphene/hexagonal boron nitride. Nat. Phys. 12, 1111–1115 (2016)

4. Y.H. Lee et al., Synthesis of large-area MoS2 atomic layers with chemical vapor deposition.

Adv. Mater. 24, 2320–2325 (2012)

5. Y.H. Lee et al., Synthesis and transfer of single-layer transition metal disulfides on diverse

surfaces. Nano Lett. 13, 1852–1857 (2013)

6. S. Wachter, D.K. Polyushkin, O. Bethge, T. Mueller, A microprocessor based on a

two-dimensional semiconductor. Nat. Commun. 8, 14948 (2017)

7. K.F. Mak, C. Lee, J. Hone, J. Shan, T.F. Heinz, Atomically thin MoS2: a new direct-gap

semiconductor. Phys. Rev. Lett. 105, 136805 (2010)

8. A. Splendiani et al., Emerging photoluminescence in monolayer MoS2. Nano Lett. 10,

1271–1275 (2010)

9. D. Xiao, G.B. Liu, W. Feng, X. Xu, W. Yao, Coupled spin and valley physics in monolayers of

MoS2 and other group-VI dichalcogenides. Phys. Rev. Lett. 108, 196802 (2012)

10. G.B. Liu, W.Y. Shan, Y.G. Yao, W. Yao, D. Xiao, Three-band tight-binding model for

monolayers of group-VIB transition metal dichalcogenides. Phys. Rev. B 88, 085433 (2013)

10 1 Introduction

11. G.B. Liu, D. Xiao, Y. Yao, X. Xu, W. Yao, Electronic structures and theoretical modelling of

two-dimensional group-VIB transition metal dichalcogenides. Chem. Soc. Rev. 44,

2643–2663 (2015)

12. K. Kosmider, J.W. Gonzalez, J. Fernandez-Rossier, Large spin splitting in the conduction band

of transition metal dichalcogenide monolayers. Phys. Rev. B 88, 245436 (2013)

13. X.X. Zhang, Y. You, S.Y. Zhao, T.F. Heinz, Experimental evidence for dark excitons in

monolayer WSe2. Phys. Rev. Lett. 115, 257403 (2015)

14. K.F. Mak, K.L. He, J. Shan, T.F. Heinz, Control of valley polarization in monolayer MoS2 by

optical helicity. Nat. Nanotechnol. 7, 494–498 (2012)

15. H.L. Zeng, J.F. Dai, W. Yao, D. Xiao, X.D. Cui, Valley polarization in MoS2 monolayers by

optical pumping. Nat. Nanotechnol. 7, 490–493 (2012)

16. T. Cao et al., Valley-selective circular dichroism of monolayer molybdenum disulphide. Nat.

Commun. 3, 887 (2012)

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References 11

Chapter 2

Time-Resolved Absorption Spectroscopy

This dissertation work emphasizes on the exciton energy shift that is induced by

femtosecond light pulses. At equilibrium, the exciton energy level in monolayer

TMDs sits in the visible regime, e.g., E0 ¼ 2 eV in monolayer WS2. In order to

detect this energy shift, we need a broadband probe pulse that is centered in the

visible spectrum. White light continuum generation is a nonlinear optical process

capable of producing such broadband light pulse between 500 and 700 nm

(hω¼ 1.8� 2.4 eV). In this chapter, we will discuss about time-resolved absorption

spectroscopy setup (or simply transient absorption) that utilizes white light con-

tinuum to probe the exciton energy shift in monolayer TMDs. In the first section, we

will show the overview of the transient absorption setup. In later sections, we will

discuss in more details about (1) laser amplifier, (2) optical parametric amplifier,



13

and (3) white light continuum generation. Finally, we will discuss about the data

analysis based on the optical physics of monolayer materials on a transparent

substrate.

2.1 Experimental Setup

2.1.1 Overview

Transient absorption spectroscopy consists of synchronized pump pulses and

broadband probe pulses to monitor the pump-induced absorption of materials.

Light has three properties, namely, photon energy, polarization, and intensity,

that, when tuned precisely, can induce various nonequilibrium phenomena in

materials. Examples are resonant excitation of quasiparticles, off-resonance

shifting of energy levels, and valley-selective light-matter interaction. For this

reason, a transient absorption setup can incorporate different stages of light man-

agement area on the optical table to generate light pulses with different photon

energies and to control their polarizations and intensities before reaching the

sample.

The overview of this transient absorption setup is schematically drawn in

Fig. 2.1. In this setup, we use a Ti/sapphire laser amplifier producing laser pulses

with duration of 50 fs and at 30 kHz repetition rate. Each pulse is split into two

arms. For the pump arm, the pulses are sent to an optical parametric amplifier

(OPA) to generate tunable photon energies and modulated by an optical chopper,

while for the probe arm, the pulses are sent through a delay stage and a white light

continuum generator. The pump and probe polarizations are varied separately by

two sets of polarizers and quarter-wave plates, allowing us to perform polarization-

dependent measurements, and an additional half-wave plate for tuning the pump

pulse intensity. The two beams are focused at the sample, and the probe beam is

reflected to a monochromator and a photodiode for lock-in detection. The optical

chopper is externally triggered at 7.5 kHz by the 30 kHz laser amplifier. Here, in

every optical modulation cycle, two adjacent pump pulses are passed, and the next

two adjacent pump pulses are blocked by the optical chopper. By scanning the

grating and the delay stage, we are able to measure induced reflection ΔR/R (hence

induced absorption Δα) as a function of energy and time delay.

2.1.2 Laser Amplifier

One important aspect to consider when setting up a transient absorption setup is the

laser pulse energy. White light continuum generation is a nonlinear optical process

that requires femtosecond pulse energy of 3–5 μJ when using a transparent

nonlinear medium such as a sapphire glass. Although this requirement is easily

14 2 Time-Resolved Absorption Spectroscopy

managed using a tiny fraction from a laser amplifier’s output, it certainly is a huge

energy barrier for a laser oscillator. Femtosecond laser amplifiers can produce pulse

energy between 10 μJ and 100 mJ, while laser oscillators are limited with pulse

energy between 10 and 100 nJ. There is an alternative to generate white light

continuum from laser oscillators by using nonlinear photonic crystal fibers. How-

ever, laser amplifiers offer more versatile options for transient absorption setup

because the remaining pulse energy can be used to generate laser pulses at various

wavelengths in the ultraviolet, visible, and mid-infrared using an optical parametric

amplifier with typical pulse energy input of 200 uJ.

In our experiment, we use a Ti/sapphire laser amplifier Wyvern 500 from KM

Labs and set its repetition rate to 30 kHz. The overview of this amplifier is

schematically drawn in Fig. 2.2 and briefly discussed as follows. A mode-locked

femtosecond oscillator Griffin (KM Labs) generates laser pulses with pulse duration

50 fs, energy 2 nJ, center wavelength 790 nm, and repetition rate 80 MHz. This

oscillator output (seed pulse) is stretched spectrally and temporally to several tens

of picoseconds using a pair of diffraction gratings before sent to pulse selection and

amplification in gain medium. This technique is called chirped pulse amplification

(CPA), and it reduces the peak intensity of the seed pulse to avoid damage to the

gain medium through self-focusing and to avoid gain saturation that prevents

further amplification. The amplifier uses a cryogenically cooled (�180 �C) gainmedium Ti/sapphire single crystal that is optically pumped to achieve population

inversion using two external pump laser diodes (Photonics Industries) each with

Fig. 2.1 Optical layout of transient absorption spectroscopy setup

2.1 Experimental Setup 15

pulse duration 20 ns (quasi-CW), power 35 W, wavelength 532 nm, and repetition

rate 30 kHz (tunable up to 300 kHz).

Regenerative amplifier cavity – The seed polarization is initially perpendicular

to the plane of incidence until it passes through an entrance Pockels cell (PC1) to

rotate the polarization by π/2 when activated. The Ti/sapphire crystal is cut at

Brewster angle so that a seed pulse that is polarized along the plane of incidence

gets trapped in the cavity and amplified. After six to ten passes, the seed pulse is

amplified by a factor of 106, and an exit Pockels cell (PC2) is activated to rotate the

polarization by π/2 so that the amplified pulse gets reflected from the Brewster’s cutsurface of the Ti/sapphire crystal and leaves the cavity. The timing of PC1 and PC2

is synchronized with the external laser diodes at 30 kHz, with 80 MHz clock from

the oscillator, such that in the time span of 33 μs, only one seed pulse is trapped andamplified.

After exiting the cavity, the amplified pulse is compressed spectrally and

temporally using (effectively) a pair of gratings to achieve short pulse duration

and very high peak intensity. The Wyvern amplifier output produces pulse duration

50 fs, energy 400 μJ (12 W at best), center wavelength 790 nm, and repetition rate

30 kHz. The laser beam is split by a thin polarizing beam splitter: (1) 230 μJ (7 W)

to pump an optical parametric amplifier and (2) 170 μJ is directed toward white

light continuum generation (3–5 μJ) where the excess energy is sent to a beam

dumper.

2.1.3 Optical Parametric Amplifier

An optical parametric amplifier (OPA) is an optical setup that converts the laser

fundamental wavelength into variable wavelengths through parametric amplifica-

tion process. The principle operation of OPA is based on an optical parametric

generation process ω0¼ωs +ωi where one photon of frequency ω0 (fundamental) is

converted into two photons of frequencies ωs (signal) and ωi (idler). The output

frequencies are varied by tuning the phase-matching condition and time delay

between pulses in a nonlinear crystal. In our experiment, we use OPA Palitra

from Quantronix that allows tunable wavelengths of signal and idler between 1.1

Fig. 2.2 Optical layout inside Wyvern laser amplifier


and 2.5 μm. Longer wavelengths up to 20 μm can be generated inside the OPA

through a difference frequency generation (DFG) ωs�ωi, while shorter wave-

lengths down to 550 nm can be generated outside the OPA through a second

harmonic generation (SHG) 2ωs or 2ωi.

The overview of this OPA is schematically drawn in Fig. 2.3 and briefly

discussed as follows. An input laser beam at wavelength 790 nm and pulse energy

230 μJ is split into two beams. About 10% of this beam will be used for the first

amplification stage and 90% for the second amplification stage.

In the first amplification stage, the 10% beam is further split into two beams. One

beam is used to generate white light continuum in a sapphire glass and directed

through a ZnSe substrate to pass only the infrared wing of the continuum and to

stretch the beam spectrally and temporally (chirped). This infrared continuum

contains the signal frequency component ωs, and it will be used as a seed pulse.

The other beam of frequency ω0 is directed to a delay stage and used as the pump

pulse. The two beams meet at the first nonlinear crystal (Crystal 1) where the optical

parametric generation ω0¼ωs +ωi occurs. The output signal and idler frequencies

can be tuned by varying the pump pulse time delay (Delay 1) to pick a particular

frequency component ωs from the seed pulse and by rotating the crystal optical axis

to achieve a phase-matching condition. The first amplification stage typically pro-

duces total pulse energy of 1 μJ (signal and idler).

In the second amplification stage, the amplified beam is directed to the second

nonlinear crystal (Crystal 2) and meets the 90% pump beam. Similarly, the ampli-

fication efficiency is tuned by varying the second pump delay (Delay 2) and by

rotating the crystal axis. This OPA is capable of producing total output energy of

45 μJ (signal + idler) with efficiency of 20% when the idler wavelength is set to

2.1 μm. The output of this OPA is used as the pump pulse in transient absorption

setup because it can provide a tunable pump wavelength at high peak intensity.

2.1.4 White Light Continuum

White light continuum generation is a nonlinear optical process that converts laser

pulses with narrow spectral bandwidth into pulses with very broad spectral

Fig. 2.3 Optical layout inside optical parametric amplifier (OPA)


bandwidth. For instance, in our setup we use a near infrared laser pulse that is

focused into a transparent nonlinear medium to generate a visible white beam when

projected on the screen (Fig. 2.4). This phenomenon was first observed by

R. Alfano and S. Shapiro in 1970 [1]. Such a spectral broadening is driven by the

nonlinear effects of self-phase modulation (SPM). This results from the Kerr

nonlinearity of materials that also gives rise to Kerr self-focusing. We discuss

this briefly as follows. Laser pulses with a Gaussian intensity time profile are

given by

I tð Þ ¼ I0exp �t2

τ2

� �ð2:1Þ

where I0 is the peak intensity and τ is the 1/e pulse duration (Fig. 2.5a). At high

intensity, laser pulses that are traveling inside a medium can induce a refractive

index change

n Ið Þ ¼ n0 þ n2I ð2:2Þwhere n0 is the linear refractive index and n2 is the nonlinear refractive index due toχ(3) nonlinearity of the medium. The electric field of an incident laser pulse

propagating in vacuum in z direction can be expressed as E(t)¼ E0 cos(kz�ωt).The time-varying laser intensity results in a time-varying refractive index of

material n(t), leading to a phase shift of the pulse’s electric field

ϕ tð Þ ¼ ω0t� kz ¼ ω0t� 2πn tð Þλ0

L ð2:3Þ

where ω0 and λ0 are the carrier frequency and vacuum wavelength of the pulse, and

L is the distance the pulse has traveled. The time-varying phase shift results in a

time-varying frequency shift of the pulse (Fig. 2.5b), and the instantaneous fre-

quency is given by

Fig. 2.4 Real image of white light continuum generated in our setup


ω tð Þ ¼ dϕ tð Þdt

¼ ω0 þ 4πLn2I0λ0τ2

� �texp �t2

τ2

� �ð2:4Þ

We can analyze the frequency shifts in three different time segments. The leading

edge of the pulse shifts to lower frequencies (red-shifted), the trailing edge shifts to

higher frequencies (blue-shifted), while the center exhibits no frequency shift.

Although the carrier frequency ω(t) varies in time, the envelope temporal profile

is effectively unchanged. Thus the spectral broadening maintains the incident pulse

duration. However, the effect of group velocity dispersion through a transparent

medium still occurs. In medium with normal dispersion, the red leading edge of the

pulse travels with a faster group velocity, while the blue trailing edge with a slower

group velocity. This effect spreads the frequency components temporally, an effect

known as chirp, which can be as wide as 1 picosecond for wavelength range of

500–700 nm (Fig. 2.6). Nevertheless, the individual packets of frequency compo-

nents still maintain the femtosecond pulse duration of the incident pulse, which is

an important criterion for ultrafast transient absorption experiment.

In the above discussion, we have shown that self-phase modulation in the time

domain can result in a white light continuum generation. Meanwhile, self-phase

modulation in the spatial domain can also result in a distinct appearance of the

white light pattern on the screen. The continuum appears as a round white disk

surrounded by a concentric rainbow-like pattern (Fig. 2.7a–d). This is called a

conical emission and it exhibits a large divergence angle. This is similar to the Kerr

self-focusing effect except now we have multiple different wavelengths in the

continuum.

Fig. 2.5 (a) Laser pulsewith a Gaussian temporal

profile. (b) Frequency shift

due to self-phase

modulation after passing

through a transparent

nonlinear medium


2.2 Data Analysis

Throughout our analysis, we used the proper definitions of reflectance R, transmit-

tance T, and absorptance α, which, respectively, means the fractions of incident

electromagnetic power that is reflected, transmitted, and absorbed at the monolayer

interface (between vacuum and substrate). This is in contrast to reflectivity and

transmittivity, which are technically only valid for semi-infinite system. We also

used more familiar names such as absorbance and absorption to mean, quantita-

tively, the absorptance α of monolayer TMDs.

Pump-probe experiments detect small changes in probe reflectance

(or transmittance) that is induced by pump excitation. This gives the differential

reflectance ΔR/R as a function of energy and time delay, from which we can obtain

the transient reflectance, R(t)¼R0(1 +ΔR(t)/R0), where R0 is the reflectance of the

system in equilibrium. In fact, the absorptance α (or the induced absorptance Δα) iswhat we really want (as shown in the main text) because it provides the explicit

information about the optical transition matrix element of the system. The absorp-

tance and the reflectance are related through the complex dielectric functioneE. Thisrelation can be derived using Maxwell equations shown in Sect. 2.2. We obtaineE ω; tð Þ by fitting R(ω, t) using a Kramers-Kronig (KK) constrained variational

analysis [2] shown in Sect. 2.1. Finally, we construct α(ω, t) by repeating this

Fig. 2.6 Incoming near IR laser pulse undergoes a self-phase modulation inside a sapphire glass,

resulting in an outgoing white light continuum with a linear chirp

Fig. 2.7 Capturing the conical emission feature. (a) Full intensity white light continuum. (b)Reduced intensity white light continuum, showing the rainbow-like pattern due to conical emis-

sion. (c) Orange color component after using 590 nm band pass filter. (d) Red color component

after using 660 nm band pass filter


procedure at different time delays. The details of the above procedure are described

as follows, in the case for monolayer TMD WS2.

2.2.1 Kramers-Kronig Analysis

First, we want to find the relation between the complex dielectric function and the

optical properties such as reflectance, transmittance, and absorptance by using

Maxwell’s equations. It is important to include the substrate influence on electro-

magnetic radiation especially for atomically thin materials. Here, the current

density in a monolayer WS2 sample is described by a delta function, jx ¼ eσ ωð Þδ zð ÞEx where eσ is the complex conductivity and Ex is the x-component of the probe

electric field (along the sample’s surface). By substituting this into Maxwell’sequations and using the appropriate boundary conditions between the monolayer

and the substrate, we can obtain the reflectance as

R ωð Þ ¼ 1� ns � ωdc E2

� �2 þ ωdc E1 � 1ð Þ� �2

1þ ns þ ωdc E2

� �2 þ ωdc E1 � 1ð Þ� �2 ð2:5Þ

and the transmittance as

T ωð Þ ¼ 4ns

1þ ns þ ωdc E2

� �2 þ ωdc E1 � 1ð Þ� �2 ð2:6Þ

where ns is the substrate’s refractive index (1.7675 for sapphire at photon energy of2.07 eV), d is the effective thickness of the monolayer (0.67 nm), and E1 and E2 arethe real and imaginary parts of the dielectric function, respectively. Here, the 2D

dielectric function is expressed as

eE ωð Þ ¼ 1þ 4πieσ=dω

ð2:7Þ

Meanwhile, the absorptance can be expressed as

α ωð Þ ¼ 4 ωdc E2

1þ ns þ ωdc E2

� �2 þ ωdc E1 � 1ð Þ� �2 ð2:8Þ

These expressions are exact, and they are valid for any monolayer materials on a

dielectric substrate. We find that the presence of the substrate significantly influ-

ences the optical properties of the monolayer WS2 above it. As compared to an

isolated monolayer WS2, the reflectance is enhanced, while both the transmittance

and the absorptance are reduced. In graphene, the above expressions can be further

simplified because the real part of its dielectric function is featureless in the visible

spectrum (E1 ~ 1, negligible σ2). This is, however, not the case for monolayer WS2,

2.2 Data Analysis 21

and we must include both the real and imaginary parts of the dielectric function to

obtain accurate results. In situation where none of the equilibrium absorptance,

reflectance, or transmittance spectrum is available, the pump-induced absorptance

Δα can still be estimated from the measured ΔR/R or ΔT/T through the following

expression:

ΔRR

¼ ns þ 1

ns � 1

� �þ ns

ns � 1ð Þ2γ21 þ γ22� �

γ2

" #Δα ð2:9Þ

ΔTT

¼ � ns þ 1

2

� �þ γ21 þ γ22� �

4γ2

� �Δα ð2:10Þ

where ΔR and ΔT are the pump-induced changes of the probe reflectance R and

transmittance T, ns is the refractive index of the substrate, γ1¼ωd(E1� 1)/c, andγ2¼ωdE2/c. In situation where γ21 and γ

22 are small, such as graphene and few other

TMDs, only the first term in the bracket needs be considered.

In our analysis, we used the equilibrium absorptance α of monolayer WS2measured using differential reflectance microscopy. The absorptance spectrum

contains peaks from the A exciton at 2.0 eV. The equilibrium reflectance R0 can

then be constructed from α by finding the appropriate complex dielectric functioneEas expressed in Eqs. 2.7 and 2.8. To do this, we implemented a Kramers-Kronig

(KK) constrained variational analysis [2] to extract ~E from the measured α in thin-

film approximation. Here, the total dielectric function is constructed by many

Drude-Lorentz oscillators, which are anchored at equidistant energy spacing, in

the following form

eE ωð Þ ¼ E1 þXNk¼1

ω2p,k

ω20,k � ω2 � iωγk

ð2:11Þ

In our calculations, we usedN¼ 40 oscillators with a fixed linewidth of γk¼ 50meV

spanning the energy range of 1.77 eV�ω0, k� 2.40 eV, and we found that these

parameters can fit α spectrum very well. We can then construct R0 spectrum by

using eE obtained from the above analysis.

The transient absorptance spectra α(t) can be obtained by performing similar

(KK) analysis. This time we inferred the absorptance from the reflectance at

different time delays: R(t)¼R0(1 +ΔR(t)/R0), where the differential reflectance

ΔR(t)/R0 is measured directly from the experiments.

2.2.2 Maxwell’s Equations for Monolayer Sample

In this section, we provide a full derivation from Maxwell’s equations in order to

obtain the exact solutions of reflectance R(ω), transmittance T(ω), and absorptance

spectra α(ω) for any monolayer materials on a substrate. Readers who are interested


in this section should also refer to the original articles by L. A. Falkovsky [3] and

T. Stauber [4] in the study of graphene. Here, we express the solutions in terms of

the complex dielectric function eE ωð Þ or conductivity eσ ωð Þ for Kramers-Kronig

analysis shown in previous section.

The electromagnetic wave equation at frequency ω, inside a medium with a

dielectric constant E and a current density j, can be expressed as

∇ ∇ � Eð Þ �∇2E ¼ Eω2

c2Eþ 4πiω

c2j ð2:12Þ

We consider a situation (Fig. 2.8) where the monolayer medium spreads on the xyplane with a current density jx ¼ eσ ωð Þδ zð ÞEx x; tð Þ that is driven by a propagating

electric field on the xz plane of the form E¼ (E0x, 0,E0z)ei(k � r�ωt). By evaluating

the partial derivatives of E, the two components of the wave equation can be

expressed as

ikx∂Ez

∂z� ∂2

Ex

∂z2� E

ω2

c2Ex ¼ 4πiω

c2jx ð2:13Þ

ikx∂Ex

∂zþ k2x � E

ω2

c2

� �Ez ¼ 0 ð2:14Þ

where we have used ∂/∂x! ikx because the law of refraction requires that kx is

conserved, while Kz is not. Boundary conditions for the tangential and normal

components of the field (red arrows) yield

Ex ¼ Ei � Erð Þ cos θi ¼ Et cos θt ð2:15Þ

Fig. 2.8 Schematic of the

plane of incidence

2.2 Data Analysis 23

EsEzjz¼0þ � Ezjz¼0� ¼ 4π

ð0þ0�

ρ zð Þdz ð2:16Þ

where we have used E ¼ Es for the substrate and E ¼ 1 for the vacuum. The charge

density ρ and the current density jx must satisfy the continuity equation ∂ρ/∂t+∇ � j¼ 0. Since they are driven by the same external field Ex(x, t), we can then

obtain a relation

ρ ¼ jxkx=ω ð2:17ÞEquation 2.16 can now be evaluated by substituting Ez from Eq. 2.14 and ρ from

Eq. 2.17, which yields

Esk2s

∂Ex

∂zþ� 1

k2z

∂Ex

∂z�¼ 4πeσ

iωExjz¼0 ð2:18Þ

where the relations between kx, kz, and ks are shown in Fig. 2.8. Note that the fieldsat the boundary areExjzþ ¼ Ete

i kxxþkszð Þ cos θt andExjz� ¼ Eieik�r � Ere

�ik�r� �cos θi.

Substituting these will yield

Esks

þ 4πeσω

� �Et cos θt ¼ 1

kzEi þ Erð Þ cos θi ð2:19Þ

Ei � Erð Þ cos θi ¼ Et cos θt ð2:20ÞThese are the two equations that will be used to obtain the reflectance,

transmittivity, and absorptance of the monolayer. For convenience, we have

moved Eq. 2.15 into 2.20.

At normal incidence, ks ¼ ffiffiffiffiEs

pω=c and kz¼ω/c; hence the coefficients of

amplitude reflection and transmission [5] can be simplified into

�r ¼ �Er

Ei

¼ 1� ns � 4πeσc

1þ ns þ 4πeσc

ð2:21Þ

t ¼ Et

Ei

¼ 2

1þ ns þ 4πeσc

ð2:22Þ

where eσ ¼ σ1 þ iσ2 is the complex conductivity of the monolayer, and we have

usedffiffiffiffiEs

p ¼ ns for an insulating substrate. Finally, we can obtain the reflectance

R and the transmittance T, as well as the absorptance α through the energy

conservation |r|2 + ns|t|2 + α¼ 1 [5],

R ¼ rj j2 ¼ 1� ns � 4πσ1c

� �2 þ 4πσ2c

� �21þ ns þ 4πσ1

c

� �2 þ 4πσ2c

� �2 ð2:23Þ


T ¼ ns tj j2 ¼ 4ns

1þ ns þ 4πσ1c

� �2 þ 4πσ2c

� �2 ð2:24Þ

α ¼ 4 4πσ1c

� �1þ ns þ 4πσ1

c

� �2 þ 4πσ2c

� �2 ð2:25Þ

The obtained α(ω) is the absorptance of a monolayer medium deposited on an

insulating substrate. The whole derivation already accounts for the out-of-phase

back-reflected electric field that reduces the light intensity impinging on the mono-

layer. The above solutions can be expressed in terms of eE instead of eσ using the

following relation:

eE ¼ 1þ 4πieσ=dω

ð2:26Þ

Note that eσ has different units in 2D (here) and in 3D; hence we keep the dielectric

function dimensionless by introducing the monolayer thickness d. In order to

convert these Gaussian-unit equations into the SI-unit, we can use 4π! 1/

E0 where E0(¼ 8.85� 10�2F/m) is the vacuum permittivity.

References

1. R.R. Alfano, S.L. Shapiro, Observation of self-phase modulation and small-scale filaments in

crystals and glasses. Phys. Rev. Lett. 24, 592–594 (1970)

2. A.B. Kuzmenko, Kramers–Kronig constrained variational analysis of optical spectra. Rev. Sci.

Instrum. 76, 083108 (2005)

3. L.A. Falkovsky, Optical properties of graphene. J. Phys. Conf. Ser. 129, 012004 (2008)

4. T. Stauber, N.M.R. Peres, A.K. Geim, Optical conductivity of graphene in the visible region of

the spectrum. Phys. Rev. B 78, 085432 (2008)

5. E. Hecht. Optics, 4th ed. (Addison-Wesley, 2002)

References 25

Chapter 3

Intervalley Biexcitons in Monolayer MoS2

Interactions between two excitons can result in the formation of bound quasiparti-

cles, known as biexcitons. Their properties are determined by the constituent

excitons, with orbital and spin states resembling those of atoms. Monolayer tran-

sition metal dichalcogenides (TMDs) present a unique system where excitons

acquire a new degree of freedom, the valley pseudospin, from which a novel

intervalley biexciton can be created. These biexcitons comprise two excitons

from different valleys, which are distinct from biexcitons in conventional semi-

conductors and have no direct analogue in atomic and molecular systems. However,

their valley properties are not accessible to traditional transport and optical mea-

surements. Here, we report the observation of intervalley biexcitons in the mono-

layer TMD MoS2 using ultrafast pump-probe spectroscopy [1]. By applying

broadband probe pulses with different helicities, we identify two species of



27

intervalley biexcitons with large binding energies of 60 and 40 meV. In addition,

we also reveal effects beyond biexcitonic pairwise interactions in which the exciton

energy redshifts at increasing exciton densities, indicating the presence of many-

body interactions among them.

3.1 Intervalley Biexcitons

Monolayer transition metal dichalcogenides (TMDs) comprise a new class of

atomically thin semiconducting crystals in which electrons exhibit strong spin-

valley coupling that results in novel valleytronic properties when interrogated with

circularly polarized light [2]. These include valley-selective photoexcitation [3–5],

valley Hall effect [6], valley-tunable magnetic moment [7], and valley-selective

optical Stark effect [8, 9]. Unlike traditional semiconductors, the Coulomb inter-

action in monolayer TMDs is unusually strong because screening is greatly

suppressed and spatial overlap of the interaction is much larger [10]. This enhances

the stability of a variety of excitonic quasiparticles with extremely large binding

energies, including excitons [11, 12], trions [13–15], and exciton-trion

complexes [16].

In addition to excitons and trions, monolayer TMDs should also host biexcitons,

van der Waals quasiparticles formed from two neutral excitons bound by residual

Coulomb fields. Moreover, the unique spin-valley coupling of these electrons,

which also behave as massive Dirac fermions at two different valleys (Fig. 3.1a),

Fig. 3.1 (a) Schematic

band structure of the two

valleys. (b) Opticaltransitions to intervalley

biexciton state

28 3 Intervalley Biexcitons in Monolayer MoS2

offers an ideal system to form unique intervalley biexcitons that have

two-dimensional positronium-molecular-like structure. Apart from the large bind-

ing energies, they are also expected to show novel properties such as entanglement

between the pair of valley pseudospins. Thorough investigation of these properties

is crucial to assess their potential use in applications. Advanced experimental

probes are needed to uncover these unique quasiparticles that cannot be accessed

by more conventional techniques.

Transient absorption spectroscopy is ideally suited to access intervalley biexciton

states via two-step excitation (Fig. 3.1b). In this experiment, an ultrashort laser pulse

is split into two portions: the first pulse (pump) is used to create a population of

excitons, |0i! |xi, and the second pulse (probe) is used to induce a second transitionto form biexcitons, |xi! |xxi. In monolayer TMDs, there are two degenerate valleys

(K and K0) where different excitons can be created using left (σ�) and right (σ+)circularly polarized light [2–5]. In order to form intervalley biexcitons, we used a

succession of pump and probe pulses with opposite helicities. The biexcitons are

revealed as a pump-induced absorption of the probe pulse at energy slightly below the

primary exciton absorption peaks.

Here, we show that there are two intervalley biexciton species in monolayer

TMD MoS2, which we identify as AA biexcitons and AB heterobiexcitons.

We measure the binding energies of these biexcitons to be 60 and 40 meV,

respectively. Experiments using excess pump photon energy reveal the stability

of the biexcitons at high temperatures. We also investigate the effect of high

excitation densities on the excitons, which shows the presence of many-body

effects in monolayer MoS2.

3.2 Experimental Methods

In our experiments, we used a Ti/sapphire regenerative amplifier producing laser

pulses at 30 kHz, with center wavelength 785 nm and duration 50 fs FWHM. Each

pulse was split into two arms. For the pump arm, the pulses were frequency

converted using an optical parametric amplifier (for resonant excitation) or a

second-harmonic crystal (for nonresonant excitation) and then chopped at

7.5 kHz. For the probe arm, the pulses were sent through a delay stage and a

white light continuum generator (hv ¼ 1.78–2.48 eV, chirp-corrected). The two

beams were focused onto the sample with 450 μm (pump) and 150 μm (probe)

FWHM diameters. The probe beam was reflected or transmitted from the sample to

a monochromator with FWHM resolution 1 nm and a photodiode for detection.

Lock-in detection at 7.5 kHz allowed measurement of fractional changes in reflec-

tance ΔR/R or transmittance ΔT/T as small as 10�4. By scanning the grating and the

delay stage, we were able to measureΔR/R orΔT/T as a function of energy and time

delay Δt, from which the induced absorptance α was obtained using Kramers-

Kronig analysis (Chap. 2). The pump fluence was varied by a combination of a half-

wave plate and polarizer, allowing us to tune the exciton excitation density. High-

3.2 Experimental Methods 29

quality monolayers of MoS2 were CVD-grown on a sapphire substrate [17, 18] and

mounted inside a cold-finger cryostat with temperature of 10 K or all measurements

in this study.

3.3 Experimental Results and Discussions

The equilibrium absorption spectrum of monolayer MoS2, measured using differ-

ential reflectance microscopy, consists of two exciton resonances (EA ¼ 1.93 eV

and EB ¼ 2.08 eV) and a background from higher-energy states [11] (Fig. 3.2a). To

create a population of excitons at the K valley, we used σ� pump pulses with photon

energy tuned near the A exciton resonance (hv ¼ 1.91 eV) and fluence 5 μJ cm�2.

Further excitation to the intervalley biexciton state can be detected from induced

absorption (Δα > 0) of σ+ probe pulses. In an ideal system with negligible

scattering, one would also expect to see optical bleaching (Δα< 0) at the A exciton

Fig. 3.2 (a) Measured

equilibrium absorbance of

monolayer MoS2. (b)Pump-induced absorption

of the probe pulses at Δt ¼0.3 ps using different

helicities. (c) The differencebetween the two Δα spectra

shown in (b), where thebiexciton binding energies

are obtained. (d) Δα spectra

at Δt ¼ 4 ps


transition using σ� probe pulses and no bleaching anywhere using σ+ probe pulses.Figure 3.2b shows a pair of Δα spectra measured using broadband probe pulses with

helicities σ� (black) and σ+ (red) at Δt ¼ 0.3 ps. This time delay was chosen to

avoid contaminating effects of coherent light-matter interaction [8, 9]. In contrast to

what is expected, we observed strong bleaching peaks not only at A but also at B

exciton transitions, and strikingly, these two peaks were present in both spectra

measured using different helicities [19].

3.3.1 Intravalley and Intervalley Scattering

The unexpected bleaching of the B exciton transition can only originate from

electron state filling in the conduction band. This is because the pump photon

energy is insufficient to excite holes to form a B exciton, and hole scattering

between the A and B bands is very unlikely due to the large energy splitting

(150 meV). Meanwhile, the photoexcited electron spin-up state (for A exciton) at

the K valley can exhibit an intravalley spin reversal to occupy the electron spin-

down state (for B exciton). This process can be mediated by flexural phonons [20],

and it can occur during the pump pulse duration (Supplementary Sect. 3.1).

Intravalley Electron Spin Reversal In order to further understand the bleaching of

B exciton transition, we note that at the K valley, the electron states in the

conduction band are split by the spin-orbit coupling into two spin states, where

the electron spin-up state (for A exciton) is 3 meV lower in energy than the electron

spin-down state (for B exciton) [21–24]. These two states preserve the good spin

quantum numbers (out of plane) due to the σh mirror symmetry of the lattice.

Resonant excitation of the A exciton using σ� pump pulses only populates the

electron spin-up state at K valley (Fig. 3.1a) and should in principle not bleach the B

exciton transition. Hence, the observed bleaching of B exciton can only be

explained by the electron spin reversal in the conduction band that immediately

occurs during the pump pulse duration (160 fs). In this flexible material, intravalley

electron spin reversal can be mediated by flexural phonons in the realm of Elliott-

Yafet spin-flip mechanism [20]. Spin-flip transition is allowed in the scattering via

long-wavelength in-plane optical phonon and out-of-plane acoustic phonon, for

which the respective deformation potentials are even and odd with respect to mirror

symmetry. Because of the small spin splitting in the conduction band, the strong

spin-orbit coupling and the flexible lattice structure, the spin lifetime could be as

short as 50 fs for suspended monolayer MoS2 at room temperature. Supporting the

membrane with sapphire substrate at 10 K can in principle prolong the spin lifetime

[20], but substrate roughness, domain boundaries, and impurities could conversely

increase the rate of carrier collisions that will enhance the spin-reversal scattering

rate. The fast electron spin reversal could thus explain the B exciton bleaching

shown in Fig. 3.2b.

3.3 Experimental Results and Discussions 31

On the other hand, the bleaching peaks that are observed using the opposite

probe helicity can be explained by intervalley scattering, which is expected to be

very fast due to electron-hole exchange interaction in this material [19, 25, 26]. In

particular, valley excitons with finite in-plane momentum can have exchange

interaction that generates an in-plane effective magnetic field, around which the

exciton pseudospin precesses from K to K0 valley. Calculations show that such

valley depolarization can be as fast as tens to hundreds of fs [25], consistent with

our observation.

3.3.2 Signature of Intervalley Biexcitons

Comparison of the two spectra in Fig. 3.2b shows that the red curve gains an

appreciable offset in certain spectral ranges with respect to the black curve that,

according to the pump-probe scheme in Fig. 3.1b, can be attributed to induced

absorption to intervalley biexciton states. To isolate the biexciton contribution, we

evaluate the difference between the two spectra (Fig. 3.2c) from which we extract

the biexciton binding energies (Δxx). We find peak energies of ΔAA ¼ 60 meV and

ΔAB ¼ 40 meV, consistent with recently estimated values [19, 27]. Since we only

excited the A excitons at resonant excitation, we expect the biexcitons to contain at

least one A exciton. Thus, by the proximity of the biexciton peaks in Fig. 3.2c to the

exciton resonances, we deduce that the former is comprised of two A excitons

(AA biexcitons), while the latter consists of one A and one B exciton

(AB heterobiexcitons). The intervalley nature of these biexcitons is guaranteed by

our measurement protocol, where we used pump and probe pulses with opposite

helicities. Additional contributions apart from the intervalley biexcitons are elim-

inated by taking the difference in Δα spectra as shown in Fig. 3.2c. At later time

delays, both Δα spectra become nearly identical (Fig. 3.2d) because intervalley

scattering establishes balance between the exciton populations at the two valleys.

Biexciton formation can then be induced from these excitons via absorption of

probe pulses with either polarization (σ� and σ+).We measured biexciton binding energies to be greater than the thermal energy at

room temperature, Δxx > 25 meV. This is expected because the excitons in

monolayer MoS2 have large binding energies (Eb), with measured values reported

from 440 [12] to 570 meV [11]. Despite this variation in the reported values, the

obtained binding energies are consistent with theoretical models [28, 29] that

predict ΔAA¼ (0.13� 0.23) Eb in monolayer MoS2 [30]. The large biexciton

binding energies share the same origin as those of the excitons where, in the 2D

limit, quantum confinement and suppressed screening greatly enhance the Coulomb

interaction in this system [10].


3.4 Time-Resolved Cooling Process

We now turn to discuss the time evolution of biexciton formation upon photoexci-

tation using excess pump photon energy (hv ¼ 3.16 eV) and exciton excitation

density of 1.4� 1012 cm�2. In this section, we measured the differential transmit-

tance ΔT/T as a function of probe photon energy and time delay (Fig. 3.3). We plot

the corresponding Δα spectra at different time delays in Fig. 3.4a, expanded around

the A exciton resonance. The induced absorption peak exhibits a gradual shift in

energy from 1.83 eV at 0.6 ps to 1.88 eV at 14 ps, which is accompanied by a peak

sharpening. To elucidate this behavior, we plot, in Fig. 3.4b, the Δα time traces

(normalized) between energies 1.78 and 1.88 eV, as indicated by the colored dots in

Fig. 3.4a. We find that the decay at 1.78 eV is accompanied by a buildup at 1.88 eV.

The latter value is consistent with the peak position of the biexciton signature

measured in the near resonant excitation.

We interpret this observation as resulting from the exciton cooling process,

where the energy distribution of the photoexcited excitons varies with the time

delay, as depicted in Fig. 3.5. In the first panel, we show the density of exciton states

D(E) surrounding the A exciton resonance. In our experiment, the nonresonant

excitation by the pump pulse (hv > EA) imparts an excess energy of δE ~ 1 eV per

exciton. This leads to the immediate formation of a hot exciton gas (Te ~ 103 K),

where the population of highly energetic excitons f(E)D(E) at E > EA becomes

substantial. This is shown on the second panel of Fig. 3.5 (gray), including the

associated Δα spectra (red). The biexciton energy can now be expressed as

EAA ¼ EA þ δEð Þpump þ EA � Δ� δEð Þprobe ð3:1Þ

where δE is the excess exciton energy whose distribution depends on Tefds. Hence,we only need to create a probe exciton with energy lower than EA � Δ to form a

biexciton of energy EAA. This explains the low-energy tail of Δα at Δt � 1.4 ps

Fig. 3.3 ΔT/T spectra as a

function of Δt after pump

pulse excitation at

hv ¼ 3.16 eV

3.4 Time-Resolved Cooling Process 33

Fig. 3.4 (a) Δα spectra at

increasing time delays,

showing the peak

sharpening and the shifting

to higher probing energy.

(b) Time traces of Δα at

probe energies as indicated

by the colored dots in (a)

Fig. 3.5 Time evolution of the exciton energy distribution (gray) and the corresponding biexcitoninduced absorption Δα (red) through the cooling process


(Fig. 3.4a). Moreover, the presence of biexciton absorption at such an early

relaxation stage of the hot exciton gas shows that they are stable at high tempera-

tures. At later time delays (Fig. 3.5, right panels), the highly energetic excitons

gradually relax into the lowest state, via releasing energy to the lattice and the

substrate, to form a cold exciton gas. During this process, the Δα spectral weight

(red) at lower energy gradually climbs to higher energy, consistent with the

observed dynamics in Fig. 3.4b and the Δα peak shifting and sharpening in

Fig. 3.4a.

3.5 Conclusions

In conclusion, by using transient absorption spectroscopy, we have observed

intervalley biexcitons in monolayer MoS2, measured their binding energies, and

monitored their relaxation processes. Studying intervalley biexcitons could offer a

new concept of quasiparticles in solids where valley pseudospin states play signif-

icant roles, apart from the usual atom-like orbital and spin states. The large

biexciton binding energies in this material offer a promising direction to search

for higher-order bound excitons such as triexcitons and the elusive quadexcitons[31, 32] as well as their interplay with spin-valley degrees of freedom. We also

found that, while two excitons can interact and form a biexciton, the interactions

between free excitons in this system reveal a large energy redshift that obeys a

simple power-law behavior with density. This experimentally obtained relation

between the energy shift and the density, as well as the obtained biexciton binding

energies, should be used to aid future theoretical works and additional experiments

in exploring many-body physics in atomically thin materials.

References

1. E.J. Sie, A.J. Frenzel, Y.-H. Lee, J. Kong, N. Gedik, Intervalley biexcitons and many-body

effects in monolayer MoS2. Phys. Rev. B 92, 125417 (2015)








Commun. 3, 887 (2012)

6. K.F. Mak, K.L. McGill, J. Park, P.L. McEuen, Valleytronics. The valley hall effect in

MoS2 transistors. Science 344, 1489–1492 (2014)

7. S. Wu et al., Electrical tuning of valley magnetic moment through symmetry control in bilayer

MoS2. Nat. Phys. 9, 149–153 (2013)

References 35

8. E.J. Sie et al., Valley-selective optical stark effect in monolayer WS2. Nat. Mater. 14, 290–294(2015)

9. J. Kim et al., Ultrafast generation of pseudo-magnetic field for valley excitons in

WSe2 monolayers. Science 346, 1205–1208 (2014)


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photocurrent spectroscopy. Sci Rep 4, 6608 (2014)

12. H.M. Hill et al., Observation of excitonic rydberg states in monolayer MoS2 and WS2 by

photoluminescence excitation spectroscopy. Nano Lett. 15, 2992–2997 (2015)

13. K.F. Mak et al., Tightly bound trions in monolayer MoS2. Nat. Mater. 12, 207–211 (2013)

14. J.S. Ross et al., Electrical control of neutral and charged excitons in a monolayer semicon-

ductor. Nat. Commun. 4, 1474 (2013)

15. C.H. Lui et al., Trion-induced negative photoconductivity in monolayer MoS2. Phys. Rev.

Lett. 113, 166801 (2014)

16. A. Singh et al., Coherent electronic coupling in atomically thin MoSe2. Phys. Rev. Lett. 112,216804 (2014)


Adv. Mater. 24, 2320–2325 (2012)



19. C. Mai et al., Many-body effects in valleytronics: direct measurement of valley lifetimes in

single-layer MoS2. Nano Lett. 14, 202–206 (2014)

20. Y. Song, H. Dery, Transport theory of monolayer transition-metal dichalcogenides through

symmetry. Phys. Rev. Lett. 111, 026601 (2013)





23. A. Kormanyos et al., Monolayer MoS2: trigonal warping, the Γ valley, and spin-orbit coupling

effects. Phys. Rev. B 88, 045416 (2013)

24. T. Cheiwchanchamnangij, W.R.L. Lambrecht, Quasiparticle band structure calculation of

monolayer, bilayer, and bulk MoS2. Phys. Rev. B 85, 205302 (2012)

25. T. Yu, M.W. Wu, Valley depolarization due to intervalley and intravalley electron-hole

exchange interactions in monolayer MoS2. Phys. Rev. B 89, 205303 (2014)

26. C.R. Zhu et al., Exciton valley dynamics probed by Kerr rotation in WSe2 monolayers. Phys.

Rev. B 90, 161302.(R (2014)

27. J. Shang et al., Observation of excitonic fine structure in a 2D transition-metal dichalcogenide

semiconductor. ACS Nano 9, 647–655 (2015)

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28, 871–879 (1983)

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Chapter 4

Valley-Selective Optical Stark Effectin Monolayer WS2

Breaking space-time symmetries in two-dimensional crystals (2D) can dramatically

influence their macroscopic electronic properties. Monolayer transition metal

dichalcogenides (TMDs) are prime examples where the intrinsically broken crystal

inversion symmetry permits the generation of valley-selective electron populations

[1–4], even though the two valleys are energetically degenerate, locked by time-

reversal symmetry. Lifting the valley degeneracy in these materials is of great



37

interest because it would allow for valley-specific band engineering and offer

additional control in valleytronic applications. While applying a magnetic field

should in principle accomplish this task, experiments to date have observed no

valley-selective energy level shifts in fields accessible in the laboratory. Here we

show the first direct evidence of lifted valley degeneracy in the monolayer TMD

WS2 [5]. By applying intense circularly polarized light, which breaks time-reversal

symmetry, we demonstrate that the exciton level in each valley can be selectively

tuned by as much as 18 meV via the optical Stark effect. These results offer a novel

way to control valley degree of freedom and may provide a means to realize new

valley-selective Floquet topological phases [6–8] in 2D TMDs.

4.1 Optical Stark Effect

The coherent interaction between light and matter offers a means to modify and

control the energy level spectrum of a given electronic system. This interaction can

be understood using what is known as Floquet theory [9], which states that a

Hamiltonian periodic in time has quasistatic eigenstates that are evenly spaced in

units of the photon-driving energy. The simplest example of this is given by a

two-level atomic system in the presence of monochromatic light, which can be fully

described by the semiclassical Hamiltonian:

H tð Þ ¼ H0 þ pE tð Þ ð4:1Þwhere H0 is the equilibrium Hamiltonian describing a two-level atom with

eigenstates |ai and |bi, p is the electric dipole moment operator of the atom, and

E(t)¼ E0 cos 2πνt is the oscillating electric field of light with amplitude E0 andfrequency ν. The perturbation term in the Hamiltonian contains a time-dependent

factor (cos 2πνt) that enables the coherent absorption of light by |ai to form the

photon-dressed state |a + hνi and the stimulated emission of light by |bi to form

another photon-dressed state |b� hνi, as shown in Fig. 4.1b. Additional states

|a� hνi and |b+ hνi, as well as higher-order terms, also arise in this situation.

This semiclassical description is consistent with the fully quantized approach.

The series of photon-dressed states formed in this way are called the Floquet

states, which can hybridize with the equilibrium states |ai and |bi through the

electric field term E in Eq. 4.1. This hybridization often results in energy repulsion

between the equilibrium and Floquet states, the physics of which is equivalent to the

hybridization between two atomic orbitals by the Coulomb interaction to form the

bonding and antibonding molecular orbitals in quantum chemistry. The interaction

between the two states always results in a wider energy level separation, and the

magnitude of the energy repulsion becomes more substantial if the states are

energetically closer to each other. In our case, there are two such pairs of states

shown in Figs. 4.1b and 4.2a denoted as {|a+ hνi, |bi} and {|ai, |b� hνi} with

identical energy difference of Δ¼ (Eb�Ea)� hν. Through the simultaneous

38 4 Valley-Selective Optical Stark Effect in Monolayer WS2

energy repulsion of these pairs, the optical transition between states |ai and |bi isshifted to a larger energy. This phenomenon is known as the optical Stark effect

[10, 11], and the corresponding energy shift (ΔE) is given by

ΔE ¼ ℳ2ab E2� �Δ

ð4:2Þ

where ℳab is the polarization matrix element between |ai and |bi and hE2i¼ E2

0=2� �

is the time-averaged value of the electric field squared, proportional to

the light intensity. Owing to the tunability of this energy shift by changing either the

light intensity or frequency, the optical Stark effect has been routinely employed in

Fig. 4.1 (a) Energy level diagram of the static Stark effect in two-level atoms. (b) Semiclassical

and (c) quantum picture of the optical Stark effect. (d) Schematic of the three Hamiltonian terms in

describing the coherent light-matter interaction

4.1 Optical Stark Effect 39

the study of atomic physics and to facilitate the cooling of atoms below the Doppler

limit [12].

4.1.1 Semiclassical Description

For the purpose of understanding the optical Stark effect in atoms, it is often

sufficient to write down the semiclassical form of the Hamiltonian, in which light

is represented by the classical fields as external perturbation. By diagonalizing the

Hamiltonian, the atomic energy levels can be obtained, and the optical Stark effect

is apparent in the light-induced change of the energy spectrum. The creation of the

Floquet states can also be seen from the energy denominator of the shift as will be

Fig. 4.2 The optical Stark effect and its observation in monolayer WS2. (a) Energy level diagramof two-level |ai and |bi atoms showing that the equilibrium and Floquet states |a + hνi and |b� hνican hybridize, resulting in shifted energy levels. (b) Measured absorbance of monolayer WS2(black) and a hypothetical absorbance curve (dashed) that simulates the optical Stark effect. (c)The simulated change of absorption induced by the pump pulses. (d) Schematic of transient

absorption spectroscopy set-up. (e) Time trace of α, induced by pump pulses of σ� helicity,

measured using probe pulses of the same (σ�, red) and opposite (σ+, black, with offset for clarity)helicities. It shows that the optical Stark effect occurs only during the pump pulse duration

(Δt ¼ 0 ps) and when the pump and probe helicities are the same


described below. In the quantum description, on the other hand, light is represented

by quantized oscillators, from which the Floquet states are seen in the quantized

photon energy spacing in the energy spectrum. In the following, we will begin by

discussing the semiclassical picture of the interaction and later compare with the

quantum picture.

Similar to the static Stark effect, the optical Stark effect also results from

induced energy repulsion between two states. To understand this, we consider a

two-level atomic system of states |ai and |bi with respective energies of Ea and Eb

(Fig. 4.1a), and we put this atom in the presence of static electric field E. Since theatomic orbitals have a definite parity, either even or odd under spatial inversion

symmetry, the application of first-order perturbation with E keeps the energy levels

unchanged. Hence, the Stark effect in atoms emerges from the second-order

perturbation with E, inducing a hybridization between states |ai and |bi that resultsin shifted energy levels by as much as

ΔEb ¼ bh jpEjaij j2Eb � Ea

¼ ℳ2ab Ej j2

Eb � Ea

ð4:3Þ

ΔEa ¼ bh jpEjaij j2Ea � Eb

¼ �ℳ2ab Ej j2

Eb � Ea

ð4:4Þ

where p is the electric dipole moment operator of the atom, E is the static electric

field strength, andℳab is the polarization matrix element between |ai and |bi. Suchenergy shifts result in a wider separation of the energy levels, also known as the

state repulsion, with magnitude

ΔE ¼ 2ℳ2

ab Ej j2Eb � Ea

ð4:5Þ

where the magnitude is quadratic in E and is inversely proportional to the energy

separation Eb�Ea before the application of the field (Fig. 4.1a).

In the optical Stark effect, the perturbation is written as H 0 tð Þ ¼ pE tð Þ, whereE(t)¼ E0 cos 2πνt is the oscillating electric field with amplitude E0 and frequency ν.Here, we can use the standard time-dependent perturbation theory [13] to find the

shift of the energy levels:

ΔEb tð Þ ¼ H0ab tð Þexp i Eb � Eað Þt

h

� �1

ih

ð t0

H0ab t0ð Þexp � i Eb � Eað Þt0

h

� �dt0 ð4:6Þ

where ΔEa(t) has a similar expression after switching the index. It is insightful to

write the perturbation Hamiltonian as H 0 tð Þ ¼ pE0 ei2πνt þ e�i2πνt� �

=2. In this way,

we can identify the perturbation as coherent absorption and emission of light from

the atomic states. By evaluating Eq. 4.6, we can obtain the time-averaged shift of

the energy levels:

4.1 Optical Stark Effect 41

Δ�Eb ¼ 1

2

ℳ2ab E2� �

Eb � Ea � hνþ ℳ2

ab E2� �

Eb � Ea þ hν

" #ð4:7Þ

Δ�Ea ¼ �1

2

ℳ2ab E2� �

Eb � Ea � hνþ ℳ2

ab E2� �

Eb � Ea þ hν

" #ð4:8Þ

where E2� � ¼ E2

0=2 is the time-averaged value of the electric field squared. These

expressions are consistent with the case of the static electric field in Eqs. 4.3 and 4.4

in which hν! 0 and E2� �! E2

0. Furthermore, by comparing the energy denomi-

nator with the static case, we can see that the two terms in Δ�Eb correspond to the

state repulsion of |bi by two photon-dressed states. These two states originate from

the coherent absorption and emission of light from state |ai, as shown in Fig. 4.1b.

Similarly, the two terms in Δ�Ea are the state repulsion of |ai by the two photon-

dressed states of |bi. These photon-dressed states are the Floquet states that we

identified in the main text. In our experiment, hν is detuned only slightly below

Eb�Ea such that the contribution from the first term to the energy shift dominates

and that of the second term can be neglected. As a result, the energy level separation

increases by

Δ�E ¼ ℳ2ab E2� �

Eb � Ea � hνð4:9Þ

This is the semiclassical result of the optical Stark shift that we obtained for Eq. 4.2

previously. As can be seen from this expression, the energy shift is linearly

proportional to the light intensity and is inversely proportional to the light energy

detuning from the Eb�Ea transition.

4.1.2 Quantum-Mechanical Description

On the other hand, the fully quantized description of the optical Stark effect can be

described by using the standard Jaynes-Cummings model [14] where light is

represented by quantized oscillators with energy of hv. The Hamiltonian is given by

H ¼ 1

2Eb � Eað Þbσz þ hνaþa þ 1

2hωR

�bσþa þ bσ�aþ� ð4:10Þ

where the three terms correspond to the two-level atom, the photon reservoir, and

the atom-photon interactions, respectively (Fig. 4.1d), hωR is the atom-photon

coupling strength (Rabi frequency), and the Pauli matrices and the ladder operators

are given by


bσz ¼ bj i bh j � aj i ah j ð4:11Þbσ� ¼ aj i bh j ð4:12Þbσþ ¼ bj i ah j ð4:13Þa nj i ¼ ffiffiffi

np

n� 1ij ð4:14Þaþ nj i ¼ ffiffiffiffiffiffiffiffiffiffiffi

nþ 1p

nþ 1ij ð4:15Þwhere |ni is the number of photons. Due to the interaction term, this Hamiltonian

only couples states |a, n+ 1i and |b, ni, which we can use as the basis for the

Hamiltonian matrix:

H ¼ hν nþ 1

2

� �1 0

0 1

� �þ 1

2

Eb � Ea � hν hωR

ffiffiffiffiffiffiffiffiffiffiffinþ 1

phωR


p � Eb � Ea � hνð Þ� �

ð4:16Þ

This Hamiltonian matrix can be diagonalized to give the energies:

En,� ¼ hν nþ 1

2

� �� 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEb � Ea � hνð Þ2 þ hωRð Þ2 nþ 1ð Þ

qð4:17Þ

As can be seen from this expression, the Floquet states emerge naturally through the

discrete energy term in units of hv (Fig. 4.1c). Additionally, even in the absence of

external perturbation of light (n¼ 0), states |ai and |bi can still exhibit hybridizationthrough the vacuum field, for which the coupling strength is hωR [15]. The optical

Stark shift of |ai! |bi optical transition can also be evaluated, after taking a weak

field approximation in which hωR�Eb�Ea� hν, giving

ΔE ffi 1

2

n hωRð Þ2Eb � Ea � hν

ð4:18Þ

where (hωR)2 is proportional to hE2i. Thus, both the semiclassical and the quantum

description of coherent light-matter interaction give consistent picture about the

Floquet states and the energy shift due to the optical Stark effect.


The sample consists of high-quality monolayers of WS2 that were CVD-grown on

sapphire substrates [16–18], and all measurements in this study were conducted at

ambient condition (300 K, 1 atm). In our experiments, we used a Ti/sapphire

amplifier producing laser pulses with duration of 50 fs and at 30 kHz repetition

rate. Each pulse was split into two arms. For the pump arm, the pulses were sent to

an optical parametric amplifier to generate tunable photon energy below the exciton

absorption (hν<Ex), while for the probe arm, the pulses were sent through a delay

stage and a white light continuum generator (hv ¼ 1.78–2.48 eV, chirp-corrected).

The two beams were focused at the sample, and the probe beam was reflected to a

4.2 Experimental Methods 43

monochromator and a photodiode for lock-in detection [19]. By scanning the

grating and the delay stage, we were able to measure ΔR/R (and hence Δα) as afunction of energy and time delay. Here, Δα(ω, t)¼ α(ω, t)� α0(ω). The pump and

probe polarizations were varied separately by two sets of polarizers and quarter-

wave plates, allowing us to perform valley-selective measurements, and an addi-

tional half-wave plate for tuning the pump pulse intensity.

4.3 Observation of the Optical Stark Effect

Similar effects have been encountered in solid-state systems, where the electronic

states are in the form of Bloch bands that are periodic in momentum space. In the

presence of monochromatic light, the Hamiltonian of crystalline solids becomes

periodic in both space and time, which leads to the creation of Floquet-Bloch bands

that repeat in momentum and energy. Floquet-Bloch bands were very recently

observed for the first time on the surface of a topological insulator irradiated by

mid-infrared light [20]. The optical Stark effect can also occur in solids through the

interaction between photo-induced Floquet-Bloch bands and equilibrium Bloch

bands [21]. To date, this effect has only been reported in a very limited number

of materials, with Cu2O [22], GaAs [23–26], and Ge [27] semiconductors among

the few examples.

Here we report the first observation of the optical Stark effect in a monolayer

TMD WS2. The recently discovered monolayer TMDs are 2D crystalline semi-

conductors with unique spin-valley properties [1] and energetically degenerate

valleys at the K and K0 points in the Brillouin zone that are protected by time-

reversal symmetry. We demonstrate by using circularly polarized light that the

effect can be used to break the valley degeneracy and raise the exciton level at one

valley by as much as 18 meV in a controllable valley-selective manner.

In monolayer WS2, the energy of the lowest exciton state is 2.00 eV at room

temperature (Fig. 4.2b, black). In order to induce a sufficiently large energy shift of

this exciton (as simulated and exaggerated for clarity in Fig. 4.2b, dashed), we use

an optical parametric amplifier capable of generating ultrafast laser pulses with high

peak intensity and tunable photon energy (1.68–1.88 eV). To measure the energy

shift we use transient absorption spectroscopy (Fig. 4.2d), which is a powerful

technique capable of probing the resulting change in the absorption spectrumΔα, assimulated in Fig. 4.2c [19]. The unique optical selection rules of monolayer TMDs

allow for Δα at the K (or K0) valley to be measured with valley specificity by using

left (or right) circularly polarized probe light (inset in Fig. 4.2e).

To search for the optical Stark effect in WS2, we start by measuring the change

in the optical absorption Δα as a function of time delay Δt between the pump and

probe laser pulses (Fig. 4.2e). To generate the necessary Floquet-Bloch bands, we

tune the pump photon energy to 1.82 eV so that it is just below the absorption peak,

with pulse duration 160 fs at FWHM, fluence 60 uJ/cm2, and polarization σ� (left-

circularly polarized). Since the optical Stark effect is expected to shift the


absorption peak to higher energy, the probe photon energy is chosen to be 2.03 eV,

which is above the equilibrium absorption peak. Figure 4.2e shows that when σ� is

used to probe Δα(red trace), there is a sharp peak that only exists at Δt ¼ 0 ps when

the pump pulse is present. This signifies that we are sensitive to a coherent light-

matter interaction occurring between the pump pulse and the sample. When σ+ is

used to probe Δα (black trace), we observe no discernible signal above the noise

level at all time delays. This shows that probing the optical Stark effect in this

material is strongly sensitive to the selection of pump and probe helicities. Closer

examination of Δα spectrum in the range of 1.85–2.15 eV reveals a faint but

noticeable background signal that is present at both valleys (Supplementary Sect.

4.7.1). In the following discussion, we only consider results taken at Δt ¼ 0 ps for

which the background signals (BG) have also been subtracted in order to focus on

the optical Stark effect.

4.4 Valley Selectivity

Having demonstrated that the optical Stark effect can be induced in a monolayer

TMD, we now study its valley specificity and demonstrate that it can be used to lift

the valley degeneracy. Figure 4.3a shows the Δα spectra induced by σ� pump

pulses probed in a valley-selective manner by using σ� (K valley) and σ+ (K0 valley)probe helicities. When the same pump and probe helicities are used (σ�σ�, Fig. 4.3aleft panel), the spectrum displays a positive (and negative)Δα at energies higher (and

lower) than the original absorption peak, clearly indicates that the absorption peak at

the K valley is shifted to higher energy via the optical Stark effect. The spectrum of

the opposite pump and probe helicities (σ�σ+, Fig. 4.3a right panel), in contrast,

shows a negligible signal across the whole spectrum, indicating an insignificant

change to the absorption peak at the K0 valley. Figure 4.3c shows identical measure-

ments of Δα using instead a σ+ pump helicity, where it can be seen that the effect is

switched to the K0 valley. This valley-selective probing of the optical Stark shift

shows that the effect is well isolated within a particular valley determined by the

pump helicity. The magnitude of the effect in either valley can also be smoothly

tuned as the pump helicity is continuously varied from fully σ� to fully σ+ (Supple-mentary Sect. 4.7.2).

The valley-specific optical Stark effect shares the same origin with the valley-

selective photoexcitation as previously reported in this class of materials [2–

4]. Both effects arise due to valley selection rules. In monolayer TMDs, the highest

occupied states in the valence band (VB) are associated with magnetic quantum

numbersm, wherem¼�1/2 and +1/2 at K and K0 valleys, respectively, as shown inFig. 4.3b and repeated in Fig. 4.3d. Meanwhile, the lowest unoccupied states in the

conduction band (CB) consist of four quasi-degenerate states, with m ¼ �3/2 and

�1/2 at K valley, and m ¼ +3/2 and +1/2 at K0 valley. Two of these states

m¼ � 1/2 have no role in the effect, thus omitted from the figures. Coherent

absorption of light by the VB creates a Floquet-Bloch band |VB+ hνi for which

4.4 Valley Selectivity 45

the magnetic quantum numbers are added by the light helicity that carries Δm¼ � 1

(σ�) or +1(σ+). The proximity in energy between this Floquet-Bloch band and the

equilibrium CB can induce a hybridization that leads to an energy shift provided

that they have the same magnetic quantum numbers. Although this material is

known to possess a strong excitonic interaction, the physical description of the

valley selectivity still remains the same, and, for the purpose of understanding the

effect, the energy of the equilibrium CB is essentially reduced by as much as the

exciton binding energy [28]. This explains the valley-selective energy shift we

observed in our experiments. These optical selection rules can be described by

Eq. 4.2 after replacingℳab byℳv, which now represents the valley selection rules

for different laser polarizations [28]. Additional experiments (below) investigating

the dependence on the light intensity and frequency show that the measured energy

shift obeys Eq. 4.2 extremely well.

4.5 Fluence and Detuning Dependences

Figure 4.4a shows a series of Δα spectra that grow with increasing pump fluence. It

can be shown (in Supplementary Sect. 4.7.3) that the integrated Δα as a function of

energy, namely, the spectral weight transfer (SWT), is proportional to the energy

shift:

Fig. 4.3 The valley selectivity of the optical Stark effect. (a) Valley-specific α spectra

(background-subtracted) induced by σ� pump pulses probed by using σ� (K valley) and σ+(K0

valley) helicities. (b) Band diagrams with the pump-induced Floquet-Bloch bands (red dashedcurves) and their associated magnetic quantum numbers m. (c, d) By switching the pump pulse

helicity to σ+, we also measured the valley-specific α spectra (c) and show their corresponding

band diagrams (d). The optical Stark effect occurs only between states having the same magnetic

quantum numbers


ð1E0

Δα ωð Þdω ¼ A ΔE ð4:19Þ

where A is the peak absorbance of the sample. In our analysis, it is sufficient to

integrateΔα in the range of 2.00� ω� 2.18 eV because the signal vanishes beyond

this upper limit. The SWT is plotted in Fig. 4.4b (blue circles) as a function of pump

fluence, together with accompanying results measured with smaller (black) and

larger (red) laser detuning energies Δ. Not only they show a linear dependence with

fluence but they also share a common slope when plotted as a function of fluence/Δin Fig. 4.4c. This is in excellent agreement with Eq. 4.2. By obtaining the peak

absorbance A¼ 0.2 from Fig. 4.2c, we can set an energy scale for ΔE in the right

vertical axis of Fig. 4.4c, which estimates an energy shift of 18 meVmeasured at the

highest fluence. We note that, for a given fluence and energy detuning, this material

exhibits the largest optical Stark shift in any materials reported to date. We have

used Eq. 4.2 to calculate the expected energy shift by using an estimated Rabi

frequency, which agrees very well with the measured value (see Supplementary

Sect. 4.7.4).

Fig. 4.4 Fluence and detuning dependences of the optical Stark shift. (a) Fluence dependence of αspectra with fluences up to 120 μJ cm�2 measured with the same pump and probe helicities. (b)Fluence and detuning dependences of the SWT with the integration range shown in (a). (c) SWT

plotted as a function of fluence/Δ, showing that all of the data points fall along a common slope

(black line). The horizontal error bars correspond to the pump bandwidth of 43 meV at full width at

half maximum. The fitting slope (black line) and the 95% confidence band (red shading) show an

excellent agreement with the characteristic dependences of the optical Stark shift in Eq. 4.2

4.5 Fluence and Detuning Dependences 47

4.6 Proposal: Valley-Specific Floquet Topological Phasein TMDs

The ability to create valley-specific Floquet-Bloch bands in monolayer TMDs

offers a means to induce new topological phases [6–8] with valley-specific edge

states. To elucidate its valley specificity, we can ignore the excitonic interaction

without loss of generality, and we conceive a situation where the σ� detuning

energy is varied across the interband transition, as shown in Fig. 4.5a (K valley) and

4.5b (K0 valley). In these figures, the coherent absorption of light (red line) from the

valence band creates a Floquet-Bloch band close to the conduction band edge

(dashed curves) and induces hybridization that results in energy repulsion (solid

curves). When the pump detuning is set at Δ< 0, band inversion should occur at K0

valley which is accompanied by an avoided crossing away from the symmetry point

(k¼K0+ δk). This gap opening allows the creation of topological edge states (red)

[29]. Here the pump helicity breaks time-reversal symmetry, inducing chiral edge

states along the boundary of the laser-exposed region where the topological order

changes (Fig. 4.6c) [30]. We note that this description is robust when the Floquet-

Bloch band is far above the interband transition. In a situation where the photon

energy approaches the exciton resonance, careful consideration of the excitonic

effects is necessary especially within the Rabi splitting of the exciton-polariton

energy dispersion [31]. In future works, it should be possible to investigate such

Floquet chiral edge states provided that the probing signal is made insensitive to

contributions from photoexcited excitons and intervalley scattering [32, 33]. This

opens an exciting avenue, which merges Floquet-driven topological transition and

valley physics together.

Fig. 4.5 Valley-specific

Floquet topological phase.

(a, b) Hybridizationbetween the Floquet-Bloch

band and the conduction

band when Δ is varied,

which gives rise to the

avoided crossing at the K

valley (a) and the band

inversion at the K0 valley(b). The crossing straight

line (red) at the K0 valley is

the anticipated chiral edge

states due to the band

inversion


The observation of a large valley-selective optical Stark effect in monolayer

WS2 represents the first clear demonstration of broken valley degeneracy in a

monolayer TMD. This is possible because of the formation of Floquet-Bloch states

via the application of circularly polarized light, which breaks time-reversal sym-

metry and allows for the lifting of the intervalley spin degeneracy. These findings

offer additional control for valley-switching applications on a femtosecond time-

scale, as well as provide a means to generate new valley-selective topological

phases in monolayer TMDs.

4.7 Supplementary

4.7.1 Time-Resolved Spectra

The equilibrium absorption spectrum of our sample (Fig. 4.7a, black), as measured

using differential reflectance microscopy, shows an absorption peak at

E0 ¼ 2.00 eV, which is also consistent with other measurements [34]. Optical

Stark effect gives rise to an energy shift of this peak (simulated in Fig. 4.7a,

dashed), which can be measured as induced absorption Δα at slightly higher energy

(simulated in Fig. 4.7b, red shaded). ThisΔα spectrum is the signature of the optical

Stark effect in our experiment.

Fig. 4.6 Schematic of the

Floquet-driven chiral edge

state along the boundary of

the laser-exposed region

4.7 Supplementary 49

In Fig. 4.7c, we present a pair of experimentally measured Δα spectra as a

function of pump-probe time delay. Here, we used left-circularly polarized (σ�)pump pulses with photon energy of 1.82 eV (below the absorption peak), fluence of

60 uJ/cm2, and duration of 160 fs at FWHM. The probe helicity is tuned to be the

same (σ�, left panel) and opposite (σ+, right panel) to the pump helicity. On the left

panel, the spectra show a distinct feature centered within the pump pulse duration

(Δt ¼ 0 fs), which consists of positive Δα above the original absorption peak and

negative below it. On the right panel, however, the spectra show only a faint and

premature signal during the pump pulse duration, in contrast to that of the left panel.

At Δt > 200 fs, both panels share common spectra, where Δα is negative at the

absorption peak and positive below it.

Fig. 4.7 (a) Measured absorbance of monolayer WS2 (black) and a rigidly shifted one (dashed) tosimulate the optical Stark effect. (b) The simulated change of absorption induced by the pump

pulse. (c) Valley-specific Δα(ω, t) spectra measured using pump-probe pulses of the same (leftpanel) and opposite (right panel) helicities. It shows a peak shift only at the K valley (Δt ¼ 0 fs)

and common background signals from the photoexcited excitons (Δt > 200 fs)


The Δα spectra at Δt ¼ 0 fs shown on the left panel indicate that the absorption

peak is shifted to higher energy. The absence of this feature on the right panel

indicates that the shift is well isolated within only the light-driven valley. The

non-zero Δα values at Δt > 200 fs in both panels, when the pump pulse no longer

persists, are induced by photoexcited excitons, due to excitonic bleaching and

biexcitonic absorption [19, 32], and they are present in both valleys. These excitons,

however, do not contribute to the optical Stark effect and just add common

background signals in both valleys. The different signatures of Δα spectra

(Fig. 4.7c, left panel) that are induced by the pump pulse (Δt ¼ 0 fs) and by the

photoexcited excitons (Δt> 200 fs) prove that the observed valley-selective energy

shift is driven coherently by light, and not by photoexcited excitons. In order to

eliminate slight contributions from the photoexcited excitons at Δt ¼ 0 fs, we take

0.5�Δα(ω,Δt¼ 400 fs) from the right panel of Fig. 4.7c as the background

(BG) signal. This BG signal will be used in the next section.

4.7.2 Polarization-Resolved Spectra

In the following discussion, we only consider results taken at Δt ¼ 0 fs where the

optical Stark effect takes place. We present the valley-specific Δα spectra, before

(Fig. 4.8a) and after (Fig. 4.8b) BG subtraction, as a function of pump polarization.

We used σ� and σ+ probe pulses to monitor the K valley (left panels) and K0 valley(right panels), respectively. The latter figures show a clear transition of Δα value:

largest when the pump pulses have the same helicities with the probe pulses, halved

when linearly polarized, and nearly zero for opposite helicities, as is also summa-

rized in Fig. 4.9b. This results from the shifting of energy levels as a function of

pump polarization, as is schematically depicted in Fig. 4.10. This behavior arises

from the selection rule in the matrix element ℳv, which is responsible for the

previously reported valley-selective photoluminescence.

4.7.3 Obtaining Energy Shift

We can represent the absorption peak by a Gaussian lineshape. The resulting

absorption spectrum due to an energy shift of ΔE can be expressed as

α ω;ΔEð Þ ¼ Aexp � ω� ΔEð Þ22c2

!ð4:20Þ

where A is the absorption peak, c is the FWHM/2ffiffiffiffiffiffiffiffiffiffiffiffi2 ln 2

p, and h¼ 1. Figure 4.11a

shows the absorption spectra before (grey shaded) and after (red curve) the peak is


shifted by ΔE. We can then obtain the change of the absorption spectrum

(Fig. 4.11b) before and after the shift:

Δα ω;ΔEð Þ ¼ α ω;ΔEð Þ � α ωð Þ¼ � α ωð Þ � α ω� ΔEð Þ

ΔE

� �ΔE

¼ � ∂α∂ω

ΔE

¼ ω

c2α ωð ÞΔE

ð4:21Þ

where we only keep terms in the first order of ΔE. This result has been routinely

used to estimate the energy shift ΔE via optical Stark effect through measuring the

Fig. 4.8 (a) Valley-specific Δα(ω, t¼ 0) spectra measured as a function of pump polarization,

before and (b) after the background (BG) signal subtraction. These spectra show that the energy

shifts are well isolated in either valley


Fig. 4.9 (a, b) Spectral polarization cuts at K (left) and K0 (right) valleys from the corresponding

panels in Fig. 4.8 for a clearer comparison

Fig. 4.10 Band diagrams showing how the energies of the conduction band (CB) and the valenceband (VB) are shifted as a function of pump polarization


change in the absorption Δα at a particular energy ω [21]. This method, however, is

insensitive to distinguishing a peak shift from a peak broadening. Thus, it is crucial

to measure the full spectrum of Δα, which can be performed in transient absorption

spectroscopy [19], and verify if the measured Δα spectrum profile results from a

peak shift (Fig. 4.11b).

To determine the shift in the peak position, we can integrate the area of this

Δα(ω,ΔE) curve in the range of 0�ω�1, which is referred in the main text as

the spectral weight transfer (SWT), and by using Eqs. 4.20 and 4.21, it can be

evaluated as ð10

Δα ω;ΔEð Þdω¼ð10

ω

c2α ωð ÞΔE

dω

¼ ΔEc2

ð10

ωAexp �ω2

2c2

� �dω

¼ AΔEð10

dxexp �xð Þ¼ AΔE

ð4:22Þ

That is, by measuring the SWT and the absorption peak A, we can extract the energyshift ΔE due to the optical Stark effect. The measured SWT is plotted in Fig. 4.4b, c

in the main text with a new energy scale as determined by Eq. 4.22.

Fig. 4.11 (a) Absorptionspectra before (black) andafter (red) the absorptionpeak is shifted by ΔE. (b)The change of the

absorption spectrum due to

shifted energy


4.7.4 Comparison from Semiclassical Theory

In theory, the optical Stark shift is completely described by the transition dipole

moment ℳab, pump field strength E0, and pump detuning Δ, which can be calcu-

lated and compared with the data in Fig. 4.4c. The energy shift has a simple

expression (Eqs. 4.9 and 4.18) given by

ΔE ¼ hωRð Þ22Δ

ð4:23Þ

where hωR(¼ℳabE0) is the Rabi frequency. We can estimate for ℳab from the

study of quantum well semiconductors [35] given as

ℳabð Þ2 ¼ eh

2E0

� �2Eg

mc

ð4:24Þ

where E0 (¼ 2 eV) is the transition energy of the exciton, Eg(¼ 2.3 eV) is the

quasiparticle bandgap [36], and mc(¼ 0.32 m0) is the effective mass of the conduc-

tion electron [37]. This gives ℳab¼ 56 Debye for monolayer WS2 (1 Debye ¼3.3� 10�30 Cm), which is about twice the value obtained for GaAs quantum wells

[38]. For the maximum energy shift of 18 meV, as measured at Δ ¼ 180 meV, we

used pump fluence of 120 uJ/cm2 with pulse width of 160 fs (obtained from

Fig. 4.2e). This gives the peak irradiance of I¼ fluence/width¼ ε0cE02/2 from

which we can determine the field strength of E0¼ 75 MV/m and the Rabi frequency

of hωR¼ 87 meV. This calculation yields the energy shift ofΔE¼ 21 meV which is

consistent with the measured value of 18 meV from our experiments.

In comparison to the optical Stark effect, the energy shift resulted if we were

applying a static field instead of the optical field is given by

ΔE ¼ 2ℳabE0ð Þ2

E0

ð4:25Þ

As can be seen, in order to obtain the same energy shift, we need E0¼ 125 MV/m,

which is very difficult to achieve in dc. Note also that the application of a static

electric field in monolayer WS2 will not lift the intervalley degeneracy.

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Chapter 5

Intervalley Biexcitonic Optical Stark Effectin Monolayer WS2

Coherent optical driving can effectively modify the properties of electronic valleys

in transition metal dichalcogenides. Here we observe a new type of optical Stark

effect in monolayer WS2, one that is mediated by intervalley biexcitons under theblue-detuned driving with circularly polarized light [1]. We find that such helical

optical driving not only induces an exciton energy downshift at the excitation valley

but also causes an anomalous energy upshift at the opposite valley, which is

normally forbidden by the exciton selection rules but now made accessible through

the intervalley biexcitons. These findings reveal the critical, but hitherto neglected,

role of biexcitons to couple the two seemingly independent valleys and to enhance

the optical control in valleytronics.



59

5.1 Blue-Detuned Optical Stark Effect

Monolayer transition metal dichalcogenides (TMDs) host tightly bound excitons in

two degenerate but inequivalent valleys (K and K0), which can be selectively

photoexcited using left (σ�)- or right (σ+)-circularly polarized light (Fig. 5.1a)

[2–5]. The exciton energy levels can be tuned optically in a valley-selective manner

by means of the optical Stark effect [6, 7]. Prior research has demonstrated that

monolayer TMDs driven by below-resonance (red-detuned) circularly polarized

light can exhibit an upshifted exciton level, either at the K or K0 valleys dependingon the helicity, while keeping the opposite valley unchanged. This valley-specific

phenomenon arises from the exciton state repulsion by the photon-dressed

(Floquet) state in the same valley, a mechanism consistent with the optical Stark

effect in other solids [8, 9].

Despite much recent progress, a complete understanding of the optical Stark

effect in monolayer TMDs is still lacking. First, the downshift of exciton level, an

anticipated complementary effect using above-resonance (blue-detuned) optical

driving, has not been demonstrated. This is challenging because the blue-detuned

light excites real exciton population, which can easily obscure the optical Stark

effect. Secondly, when the detuning is sufficiently small and comparable to the

biexciton binding energy, the effect may involve a coherent formation of the

recently identified intervalley biexcitons [10, 11]. These biexcitons are expected

to contribute to the optical Stark effect, as indicated by earlier studies in semicon-

ductor quantum wells [12, 13]. Elucidating these processes is therefore crucial to

understand the coherent light-matter interactions in monolayer TMDs.

In this chapter, we explore the optical Stark effect in monolayer WS2 under blue-

detuned optical driving. By pumping the system above the A exciton resonance

using left-circularly polarized laser pulses, we can lower the exciton energy at the Kvalley. In addition, as the driving photon energy approaches the resonance, an

unexpected phenomenon emerges – the exciton energy at the opposite (K0) valleyis raised. This observation is anomalous because interaction with the driving

photon is forbidden at this valley by the exciton selection rules. The upshifted

Fig. 5.1 (a) K and K0 valleys couple selectively with left (σ�) and right (σ+) circularly polarized

light due to selection rules. (b) Schematic of the pump-probe spectroscopy setup

60 5 Intervalley Biexcitonic Optical Stark Effect in Monolayer WS2

exciton level also contrasts sharply with the downshifted level at the K valley.

These findings reveal the strong influence of intervalley biexcitons on the optical

Stark effect. By including their contributions in an expanded four-level scheme of

optical Stark effect, we are able to account for all the main observations in our

experiment.


High-quality monolayer WS2 samples were grown by chemical vapor deposition

(CVD) on sapphire substrates [14–16]. We performed all of the optical measure-

ments at ambient condition (300 K, 1 atm). In our experiments, we used a

Ti/sapphire amplifier that emits laser pulses with 50 fs duration and 30 kHz

repetition rate. Each pulse was split into two arms. For the pump arm, the pulses

were sent to an optical parametric amplifier to generate tunable pump photon

energy. For the probe arm, the pulses were sent through a delay stage and a white

light continuum generator (hν¼ 1.78 – 2.48 eV, chirp-corrected). The pump and

probe polarizations were varied separately by two sets of polarizers and quarter-

wave plates (Fig. 5.1b). The two beams were focused onto the sample, and the

reflected probe beam was diffracted by a monochromator and measured by a

photodiode for lock-in detection. By scanning the grating and the delay stage, we

were able to measure ΔR/R as a function of photon energy and time delay, from

which we extracted the corresponding change of absorption Δα(ω, t)¼ α(ω,t)� α0(ω) using the method in Ref. [11].

5.3 Experimental Results and Data Analysis

We monitor the pump-induced change of exciton levels at the K (K0) valley by

using the reflection of synchronized broadband probe pulses with σ� (σ+) polari-zation (Fig. 5.1b). Our samples are high-quality WS2 monolayers grown by chem-

ical vapor deposition on sapphire substrates [14–16]. The change of absorption

(Δα) in monolayer WS2 can be extracted from the change of reflection by using

Kramers-Kronig analysis within a thin-film approximation [10, 16]. The resulting

Δα spectrum typically shows a single-cycle waveform, which allows us to deter-

mine the direction and magnitude of the exciton energy shift (ΔE) (Fig. 5.2a, b). Inour experiment, we examine the lower-energy part of Δα spectrum (filled color in

Fig. 5.2a, b), because the coherent contribution is more pronounced below the

energy resonance (E0). For a blue-detuned optical Stark effect (Fig. 5.3a), the pump

photon energy lies slightly higher than the A exciton resonance in monolayer WS2,

which is at E0¼ 2.00 eV from our measured absorption spectrum (Fig. 5.3b) as well

as from other experiments [17]. We observe no side peak or shoulder below the

5.3 Experimental Results and Data Analysis 61

exciton absorption peak, implying that our samples have weak trionic effect

(Fig. 5.3b).

Figure 5.4a, b shows the Δα spectra at zero pump-probe delay at three different

pump photon energies (hν¼ 2.10, 2.07, 1.99 eV) but the same pump fluence (28 uJ/

cm2). We display the spectra in the range of 1.80–1.96 eV, where the coherent

effect is more pronounced and less contaminated by the pump scattering. For the σ�

probe (Fig. 5.4a), the spectral shape is similar to that in Fig. 5.2a, indicating a

redshift of the exciton level at the same (K) valley. As the pump photon energy

approaches the resonance from 2.10 to 1.99 eV, the magnitude increases consider-

ably, indicating an increasing redshift of the exciton level. This observation com-

plements the previous studies, which reported a blueshift under red-detuned optical

driving. In contrast, the Δα spectra at the opposite (K0) valley exhibit a distinct

Fig. 5.2 (a, b) Simulated absorption spectra α(ω) that are shifted by ΔE to lower and higher

energies (upper panels), as well as their induced absorption spectra Δα(ω) (lower panels)

Fig. 5.3 Blue-detuned optical Stark effect and its observation in monolayer WS2. (a) Blue-

detuned optical driving scheme, where we use σ� pump pulse with photon energy hν slightly

above the exciton resonance E0. (b) Measured absorption spectrum of monolayer WS2 shows that

E0 ¼ 2.00 eV at 300 K (black curve), which can be fitted with a single Lorentzian (neutral) excitonpeak plus a background polynomial slope from the higher-energy states (red curve)


form, as revealed by the σ+ probe (Fig. 5.4b). Its value is negative in the range of

1.80–1.90 eV, with a waveform similar to that in Fig. 5.2b. This indicates a

blueshift of the exciton level at the K0 valley, which becomes more substantial as

the pump approaches the resonance (see Supplementary Sect. 5.6.3 for more

discussions on both blue- and red-detuned pumping).

The observed spectra include both the coherent signals from the optical Stark

effect and the incoherent signals from the exciton population that is unavoidably

generated by the above-resonance optical pumping [10, 11, 18–25]. The coherent

signals are known to appear only within the pump pulse duration, whereas the

incoherent signals remain after the pulsed excitation. Their different time depen-

dences allow us to separate them by monitoring the temporal evolution of Δα.Figure 5.5a, b shows the time traces after the excitation with 200 fs laser pulses at

photon energy hν¼ 2.03 eV. At pump-probe delay Δt> 1 ps,Δα is similar for both

valleys, with positive value at the probe energy of 1.84 eV but negative value at

1.95 eV. These features correspond to the exciton population effects, where the

slow rise in Fig. 5.5a shares the same timescale with the intervalley scattering. This

suggests that the initial population-induced dynamics arises from the same scatter-

ing mechanism, which can be mediated by defects and electron-phonon interac-

tions. At zero pump-probe delay, however, the two valleys exhibit significantly

different response. The difference can be attributed to the optical Stark effect, a

coherent process that follows the pump pulse intensity profile. At probe energy

1.84 eV, the coherent contribution is particularly prominent and can be readily

separated from the incoherent background by direct extrapolation (insets of

Figs. 5.5a, b and 5.10 in the Supplementary Sect. 5.6.2).

We have extracted the coherent component of �Δα at 1.84 eV and plot the

values as a function of pump fluence (Fig. 5.6). The associated energy shift ΔE can

be estimated from the differential form Δα(ω,ΔE)¼ � (dα/dω)ΔE. Given the

Fig. 5.4 (a, b) Valley-specific Δα spectra induced

by σ� pump pulses

(hν ¼ 1.99, 2.07, 2.10 eV)

and monitored by using σ�

(K) and σ+ (K0) broadbandprobe pulses at pump-probe

time delay Δt ¼ 0. The

increasing Δα at K valley

indicates a pump-induced

redshift of exciton energy.

On the other hand, the

decreasing Δα at K0 valley,though unexpected,

indicates a pump-induced

blueshift of exciton energy

5.3 Experimental Results and Data Analysis 63

measured �Δα and the slope at 1.84 eV, we have evaluated such energy shift (the

right vertical axis of Fig. 5.6). Our result shows that the exciton level at K and K0

valleys, respectively, downshifts (�4 meV) and upshifts (9 meV) under the σ�

blue-detuned optical driving. The magnitude of both shifts increases sublinearly

with pump fluence, in contrast to the linear fluence dependence in prior red-detuned

experiments.

5.4 Intervalley Biexcitonic Optical Stark Effect

The upshift of exciton level at the K0 valley is anomalous. First, according to the

well-known selection rules in monolayer WS2, the K0 valley is not accessible by the

σ� (K valley) optical driving. The observed optical Stark effect at K0 valley

apparently violates this selection rule. Secondly, even if the access to the K0 valleyis allowed, a blue-detuned optical driving is expected to downshift the exciton

level, as in the case of the K valley. The energy upshift at the K0 valley apparently

defies this common knowledge of optical Stark effect; hence, it must arise from a

Fig. 5.5 (a, b) Time traces

of Δα induced by σ� pump

pulses (hν ¼ 2.03 eV) and

monitored at probe energy of

1.84 and 1.95 eV with

different helicities. The topinset shows the curve-fittingdecomposition of the coherent

and incoherent signals.

The bottom inset shows thevalley contrast of the signals,

Δα(σ+)�Δα(σ�), where thetwo curves are offset forclarity


different mechanism, one that is beyond the framework of interaction between light

and single excitons.

We attribute this phenomenon to the optical Stark effect that is mediated by

intervalley biexcitons. Recent research has revealed significant interactions

between individual excitons in monolayer TMDs. In particular, two excitons at

different valleys can be bound to form an excitonic molecule, the intervalley

biexciton, with unusually large binding energies (Δb ¼ 40–70 meV) [10, 11, 26–

28]. These intervalley biexcitons offer an effective channel to couple the two

valleys, with selection rules different from those of single excitons. In view of

such strong biexcitonic effect, we can account for our observations within a four-

level scheme, which includes the ground state j0i, the two valley exciton states jxiand jx0i, and the intervalley biexciton state jxx0i (Fig. 5.7). In this scheme, the

optical pumping creates two types of photon-dressed (Floquet) states – one from the

ground state j0 + hνi and the other from the biexciton state jxx0 � hνi. The former

can interact with the exciton state jxiat the K valley. Since j0 + hνi lies above jxiin ablue-detuned experiment, repulsion between the two states causes jxi to downshift.This is responsible for the ordinary optical Stark effect at the K valley (red dots in

Fig. 5.6). In contrast, the biexciton Floquet state jxx0 � hνi can interact with the

exciton state jx0i at the opposite (K0) valley according to different selection rules forthe intervalley biexciton. Since jxx0 � hνi lies below jx0i, repulsion between the twowill cause jx0i to upshift. This is responsible for the anomalous optical Stark effect

at the K0 valley (blue dots in Fig. 5.6). It is evident that the intervalley biexciton

plays a unique role in coupling the two valleys, and the effect can be utilized for

enhanced control of valley degree of freedom [29].

Fig. 5.6 Fluence

dependence of the blue-

detuned optical Stark shift.

The measured data of �Δα(/ΔE) are plottedat increasing pump fluence

(σ�, hν ¼ 2.03 eV),

measured valley selectively

at probe energy of 1.84 eV.

The energy scale on the

right axis is estimated based

on the measured absorption

slope of 0.2/eV at 1.84 eV.

Note that the energy scaling

is different between the two

valleys. The fitting curves

show that the K and K0

valleys exhibit square-root

fluence dependences, as

discussed in the main text

5.4 Intervalley Biexcitonic Optical Stark Effect 65

5.4.1 Four-Level Jaynes-Cummings Model

In order to investigate the photoinduced coupling between these states, we consider

a four-level Jaynes-Cummings model, with a procedure similar to but extended

from our previous work [6, 30]. Such a model has been successfully applied to

describe the light-dressed states in many semiconductor systems and can readily be

adopted to describe the exciton-biexciton system [6, 8, 9, 31–33]. By virtue of the

unique valley selection rules in this system, the originally 4� 4 Hamiltonian matrix

can be simplified into two decoupled 2� 2 Hamiltonian matrices:

HK ¼ 1

2E0bσz þ hνaþa þ 1

2g�bσþa þ bσ�aþ

� ð5:1Þ

HK0 ¼ 1

2E0 � Δbð Þbσz þ hνaþa þ 1

2g0�bσþa þ bσ�aþ

� ð5:2Þ

The three terms from left to right in each Hamiltonian correspond to the two-level

system, the photon reservoir, and the exciton-photon interactions, respectively.

Possible contribution from real exciton population is neglected in this model.

Here, g and g0are the exciton-photon coupling strengths, bσ’s are the Pauli matrices,

a and aþ are the photon ladder operators, and Δb is the biexciton binding energy.

The Hamiltonian HK couples states j0, n+ 1i and jx, ni, while HK0 couples states jx0,ni and jxx0, n� 1i, where n is the number of photons. By using these states as the

basis, we can express the Hamiltonians as the following matrices:

HK ¼ 1

2

hν� E0 gffiffiffiffiffiffiffiffiffiffiffinþ 1

pgffiffiffiffiffiffiffiffiffiffiffinþ 1

p � hν� E0ð Þ� �

ð5:3Þ

Fig. 5.7 Energy level diagram of the intervalley biexcitonic optical Stark effect. Here the σ�

pump pulse is blue-detuned above the energy resonance between the ground state j0i and the

exciton state jxi. Coherent absorption from j0i results in a photon-dressed state j0 + hνi, whilecoherent emission from the intervalley biexciton state jxx’i results in a photon-dressed state jxx0� hνi. Additional states jx� hνi and jx0 + hνi also arise in this situation, but we omit them from

this figure for the sake of clarity


HK0 ¼ 1

2

hν� E0 þ Δb g0ffiffiffin

pg0

ffiffiffin

p � hν� E0 þ Δbð Þ� �

ð5:4Þ

Here we omit the energy offsets hν n� 12

� �of the photon reservoir. Diagonalization

of the Hamiltonian matrices gives the eigenenergies of the photon-dressed states

EK ¼ �12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihν� E0ð Þ2 þ g2 nþ 1ð Þ

qand EK0 ¼ �1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihν� E0 þ Δbð Þ2 þ g02 nð Þ

q,

where gffiffiffiffiffiffiffiffiffiffiffinþ 1

p ¼ ℳE0 and g0ffiffiffin

p ¼ ℳ0E0 are the Rabi frequencies. Here ℳ and

ℳ0are the moments for |0i! jxi and jx0i! jxx0i transitions, respectively, and ε0 is

the electric field amplitude of the light. From these expressions, we can finally

obtain the optical Stark shifts of the exciton states:

ΔEK ¼ �1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihν� E0ð Þ2 þℳ2E2

0

q� hν� E0ð Þ

� �ð5:5Þ

ΔEK0 ¼ þ1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihν� E0 þ Δbð Þ2 þℳ02E2

0

q� hν� E0 þ Δbð Þ

� �ð5:6Þ

Despite much similarity, the two optical Stark effects are quantitatively different,

because the transition moments are generally different, and the biexciton photon-

dressed state is offset byΔb. In the large detuning limit hν�E0�ℳE0, we retrieve

the well-known expression ΔEK ¼ �ℳ2E20=4 hν� E0ð Þ with a linear fluence

dependence, as observed in the previous red-detuned experiment. Conversely, in

the small detuning limit hν�E0�ℳE0, we obtain ΔEK ¼ �1

2

ffiffiffiffiffiffiffiffiffiffiffiffiℳ2E2

0

qwith a

square-root fluence dependence. The observed sublinear fluence dependence in

Fig. 5.6 indicates that the small detuning limit is reached for both valleys in our

experiment. Our fluence dependence data can be fitted with this model (Fig. 5.6),

with transition moments and effective detunings as adjustable parameters

(supporting information). The good agreement between the experiment and the

model strongly supports that this optical Stark effect is mediated by intervalley

biexcitons. We note that trion states can in principle also exhibit an optical Stark

effect. However, given the small charge background with no trion absorption

feature in our samples (Fig 5.3b), we do not expect the trion states to play a

significant role in our observation.

In summary, we have observed an exciton energy downshift at the excitation

(K) valley and an energy upshift at the opposite (K0) valley, under the blue-detunedoptical driving in monolayer WS2. While the energy downshift arises from the

single-exciton optical Stark effect, the anomalous energy upshift is attributed to the

intervalley biexciton optical Stark effect because it exhibits three characteristics:

(1) It emerges only within the pump pulse duration, (2) it has a square-root

dependence on the pump fluence, and (3) it obeys the biexcitonic valley selection

rule for opposite circularly polarized light, consistent with our model. Our results

show that the intervalley biexciton is not only a rare and interesting quasiparticle by

itself, but it also plays an active role to channel a coherent and valley-controllable

light-matter interaction.

5.4 Intervalley Biexcitonic Optical Stark Effect 67

5.5 Perspective: Zeeman-Type Optical Stark Effect

Apart from slight quantitative difference, the two types of optical Stark effects

exhibit beautiful contrast and symmetry with the valley indices (K, K0) and the

direction of the energy shift (down- and upshifts). The optical Stark effect at K

valley arises from intravalley exciton-exciton interaction through statistical Pauli

repulsion, whereas the effect at K0 valley arises from intervalley exciton-exciton

interaction through biexcitonic Coulomb attraction. Altogether, the two effects

induce opposite energy shift at the two valleys, in contrast to the prior

red-detuned optical Stark effect that occurs at only one valley [6, 7]. This behavior

is analogous to the Zeeman effect, which splits antisymmetrically the electronic

valleys under applied magnetic field [34–37]. We may therefore call this new

phenomenon a Zeeman-type optical Stark effect, in which the circularly polarized

light plays the role of the magnetic field that breaks time-reversal symmetry and

lifts the valley degeneracy (Fig. 5.8). This new finding offers much insight into

coherent light-matter interactions in TMD materials and may find important appli-

cations in the design of TMD-based photonic and valleytronic devices.

5.6 Supplementary

5.6.1 Coherent and Incoherent Optical Signals

Unlike in the red-detuned experiments [6, 7], the incoherent signals in our blue-

detuned experiments are unavoidable and quite significant, because a large popu-

lation of excitons is generated by the optical pumping at above-resonance energy

(hν> E0). It is therefore important to distinguish the coherent features of the optical

Stark effect from the incoherent background due to the photoexcited excitons. Such

incoherent signal corresponds to the exciton bleaching, biexciton absorption, and

band renormalization, as shown by earlier studies [10, 11, 18–25]. These

Fig. 5.8 Zeeman-type

optical Stark effect

illustration. The red arrowrepresents a circularly

polarized light that breaks

time-reversal symmetry and

acts like an applied

magnetic field. Coherent

interaction between the

light and monolayer WS2results in a virtual creation

of magnetic field lines

(blue)


photoexcited excitons do not contribute to the optical Stark effect and just add

common background signals contributing to the final profiles of the Δα spectra in

both valleys.

We can exclude other incoherent processes as the responsible mechanism for the

observed valley-contrastive energy shifts at time zero. The population-induced

biexciton absorption, Pauli blocking, and renormalization of band structure should

not be responsible, because they normally persist as long as the exciton recombi-

nation time (>1 ps). The exciton population imbalance between the K and K0

valleys lasts for ~1 ps, still longer than the pulse duration of our pump laser

(~200 fs). But it may contribute to the slow rise (~1 ps) of valley-contrastive signal

in Fig. 5.5a, possibly due to a small renormalization of band structure. In respect to

a possible exciton-exciton interaction, excitons in the same (different) valley are

expected to repel (attract) one another and cause a blue (red) shift of the exciton

energy, in analogy to the spin-dependent interactions of excitons in semiconductor

quantum wells [38, 39]. These predicted valley-dependent energy shifts are, how-

ever, opposite to the energy shifts in our experiment. We can therefore conclude

that the valley-contrastive energy shift is a coherent phenomenon.

The different contributions to the optical signals are illustrated in Fig. 5.9a–c.

Note that this is not an attempt to fit the experimental data, which otherwise requires

additional details beyond the present study. Here, the energy shift (OSE) contribu-

tion results from the absorption difference between the shifted and the original

absorption spectrum. The bleaching contribution accounts for the population-

induced Pauli-blocking effect such that the material becomes less absorptive. In

this illustration, we use an example where 40% of the absorption peak is Pauli

blocked. Here we assume that the electron population has not rearranged them-

selves to occupy the shifted absorption peak, so that the bleaching profile is simply

derived from the original absorption peak. So, the OSE and the bleaching effects are

considered as higher-order perturbations, and their contributions can be treated

separately for the purpose of illustration. We also considered a situation where the

electron population has quickly rearranged themselves to occupy the shifted

absorption peak, and this would still result in a very similar spectral profile. The

biexciton contribution is simply taken as a copy of the original absorption peak

(here 30%) that is shifted toward lower energy by the amount of biexciton binding

energy. The resulting Δα spectral profiles can be easily reproduced with a wide

range of variation from individual contributions, which are consistent with our

measurements within the probe windows (Fig. 5.4).

We also want to note that contributions from imperfections or defects can be

ruled out in our experiments. Imperfections in monolayer TMDs are well known to

have extremely small optical absorption compared to the excitons, even under the

excitation of ultrafast pulses. According to other photoluminescence experiments,

their contribution is significant only at very low temperature when the trapping is

efficient, not at room temperature that we performed our experiment. In addition,

their contributions, if there were any, would lack the valley and coherent charac-

teristics shown in our experiment.


5.6.2 Time-Trace Fitting Decomposition Analysis

The coherent and incoherent responses can be separated by their distinct depen-

dences on the pump-probe delay. The incoherent processes evolve with the popu-

lation of excitons, which accumulates during the pump pulse duration and remains

afterwards. On the other hand, the coherent processes follow instantaneously the

pump pulse intensity profile. Figure 5.10a shows the experimentally measured Δαspectra as a function of time delay. Here, we used σ� pump pulses at hν ¼ 1.99 eV,

fluence of 28 uJ/cm2, and duration of 200 fs. The spectra at K0 valley shows a clear

coherent effect at Δt ¼ 0 that persists only as long as the pump pulse duration.

Meanwhile the coherent effect at K valley is comparable with the incoherent

effects, because photoexcitation is most effective at this valley due to the selection

rules. Figure 5.10b shows the time cuts of Δα spectra at �0.4, 0, and 0.4 ps, which

Fig. 5.9 A simple illustration for different contributions to the optical signals. Note that this is not

an attempt to fit the experimental data, which otherwise requires additional details beyond the

present study. (a) Simulated absorption spectra α(ω) that are shifted byΔE to lower (K) and higher

energies (K0). We usedΔE at K0 valley twice that of the K valley, to be consistent with the obtained

ΔE in our experiments. (b) Contributions to Δα(ω) from the optical Stark effect (OSE), exciton

bleaching, and biexciton absorption. Note that the biexciton absorption should in principle occur

only for cross-polarized pump-probe at K0 valley. In situation where intervalley scattering happensduring the pump pulse duration, the biexciton absorption can also appear at the K valley. (c) TotalΔα(ω), where the boxed region corresponds to the probe window in our experiment (Note that here

we do not include the large contaminating signal from the pump scattering)


shows that the spectra exhibit a strong valley contrast at 0 ps due to the coherent

process. At longer time delays, the coherent process vanishes and followed by the

incoherent process that will have similar magnitudes at both valleys due to the

intervalley scattering of excitons.

Figure 5.11a shows Δα time traces that are induced by hν ¼ 2.03 eV (σ�) andvalley-selectively measured with probe energy of 1.84 eV. Consistent with the

above description, we find that the time traces exhibit a strong valley contrast at

Δt ¼ 0 (Δα(σ�σ�)�Δα(σ�σ+), see Fig. 5.11b) where the coherent signal is

expected and followed by a slow dynamics with diminishing valley contrast

where the incoherent signal remains. We performed curve-fitting decomposition

in order to separate the Gaussian-like coherent signal (y1) from the slowly building-

up incoherent signal (y2) as a function of time delay (x), which can be expressed as

y1 ¼ A1exp � x� x0ð Þ2t21

!ð5:7Þ

y2 ¼ A2exp � x� x0t2

� �1� erf

�4 x� x0ð Þt2 þ s2

2ffiffiffi2

pst2

� �� ð5:8Þ

where A1 and A2 are the amplitudes, x0 is the time zero, t1 is the Gaussian pulse

width, t1 is the decay time constant, and s is the Gaussian buildup time. By using

these fitting parameters, we can fit the time traces at K and K0 valleys and obtain theamplitudes A1 and A2. As described earlier, the amplitude A2 corresponds to the

Fig. 5.10 (a) Valley-specific Δα(ω,t) spectra induced by σ� pump pulses at hν ¼ 1.99 eV and

measured using probe pulses of σ� (K) and σ+ (K0) helicities at 300 K. It shows a clear coherent

signal at K0 valley (Δt ¼ 0 fs) and incoherent signals from the photoexcited excitons afterwards.

(b) Time cut Δα spectra at �0.4, 0, 0.4 ps


incoherent processes from the photoexcited excitons, which is determined by the

total number of created excitons and proportional to the fluence. By subtracting out

this component from the data points taken at different fluences, the coherent

components can be extracted and plotted as a function of pump fluence (Fig. 5.6).

5.6.3 Possible Effects Under Red-Detuned Pumping

We remark on a possible observation of biexcitonic optical Stark effect under

red-detuned optical driving. In the biexcitonic scheme (Fig. 5.7), the K0 valley is

expected to upshift (downshift) if the red-detuning energy is smaller (larger) than

the biexciton binding energy Δb. In our experiment, however, we could not reveal

these complementary biexcitonic features due to multiple technical difficulties.

According to our observation in the blue-detuned configuration, the biexcitonic

optical Stark effect emerges only when the detuning energy is smaller than Δb. In

the red-detuned configuration with such a small detuning energy, however, the

scattering background from our pump pulses (spectral width ~35 meV) becomes

rather significant and precludes us from detecting any signals of the optical Stark

effect. Further research is merited to overcome these difficulties and reveal more

details of the biexcitonic optical Stark effect in monolayer WS2.

Fig. 5.11 (a) Time traces

of Δα induced by σ� pump

pulses (hν ¼ 2.03 eV) and

monitored at probe energy

of 1.84 eV with different

helicities. Curve-fitting

decomposition shows the

Gaussian-like coherent

signals (color shaded) andthe slow buildup of

incoherent signals (dashedlines). (b) The difference,Δα(σ�)�Δα(σ+), showsthat the valley contrast is

dominated by the coherent

signal at Δt ¼ 0


5.6.4 Fitting Analysis

The sublinear dependence in Fig. 5.6 suggests that the small detuning limit has been

reached in both valleys. In order to verify this, we performed fitting analysis for

both valleys. Note that the optical transition between the ground state and the

exciton states is affected by the shifts of the three states j0i, jxi, and jx0i, as canbe seen in Fig. 5.7. Hence, the fluence dependences of the two valleys must be

treated separately. At K valley, the j0i and jxi states are shifted symmetrically

toward each other, leading to a redshift of the optical transition. At K0 valley, thej0i and jx0i states are both shifted to a higher energy, but the magnitude of the

j0i shift is observed to be much smaller than that of the jx0i, hence can be neglected,leading to a blueshift of the optical transition.

The resulting shifts of the optical transitions can be expressed as

y1 ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21 þ b1x

q� a1

� �for K valley, and y2 ¼

1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia22 þ b2x

q� a2

� �for K0

valley, where a1, a2 and b1, b2 are fitting parameters that represent the effective

detunings and the transition matrix elements, respectively, and x is the pump

fluence. Fitting curves using these expressions show excellent agreements with

the fluence dependence data in Fig. 5.6. The normalized and dimensionless fitting

results give a1� 0, b1¼ 0.038 (�0.006) for K valley, and a2¼ 0.48 (�0.21),

b2¼ 0.202 (�0.036) for K0 valley. Note that these fitting parameters are obtained

from data in Fig. 5.6 that are normalized at each valley. Hence, their absolute values

have only limited use. Nevertheless, we can still make meaningful comparison on

their relative values within each valley, e.g., a1 vs b1 and a2 vs b2.These fitting results have two important implications. First, it suggests that the

detuning at K valley is effectively zero although the nominal detuning is 30 meV,

resulting in a perfect square-root dependence with fluence. This seeming discrep-

ancy is reasonable because, unlike in atomic systems, the energy levels in solids

form a band of resonance. Here the pump pulse has a finite bandwidth of 35 meV,

and its center pump energy is located within the exciton linewidth of about 60 meV;

hence, a quasi-resonant condition may have already occurred. Secondly, the effec-

tive detuning at K0 valley is found to be larger than that of the K valley. Neverthe-

less, the detuning term is still significantly smaller than the Rabi frequency term

a22 � b2x� �

, resulting in a majorly square-root fluence dependence. In this way, the

intervalley biexciton is found to play significant roles in increasing the effective

detuning by Δb and in enhancing the effective Rabi frequency. In fact, the effective

moments can be roughly estimated from the fitting results, yielding ℳ� 6 Debye

andℳ0 � 140 Debye, which shows that the effective Rabi frequency at K0 valley is

indeed much enhanced due to the intervalley biexcitons. This could be attributed to

the effective self-consistent local field (described below).

These findings display a rare and fascinating interplay in the coherent light-

matter interaction where now Coulomb interactions play a significant role.


Previously, the optical Stark effect has been conveniently interpreted in terms of the

well-established dressed-atom picture. This approach is proven suitable for exci-

tons in monolayer TMDs at large detuning below resonance where contribution

from Coulomb interaction is negligible [6, 7]. Despite the broad resemblance as it

may seem, however, excitons are not atoms, and their differences are most pro-

nounced at small detuning limit, in the scale of biexciton binding energy [40],

where many-body interactions in solids due to Coulomb interaction cannot be

ignored. As pointed out in Ref. [41], in addition to the applied field, the molecular

field associated with the other excitons can combine to give an effective self-

consistent local field. Therefore, it can be expected that the contribution of this

molecular field is significant when the pump photon energy is close to exciton

resonance (hν ~ E0). This gives rise to an optical Stark effect that is enhanced

through the renormalized Rabi frequency, consistent with the observed effect at K0

valley in our experiments. The situation is different from the far red-detuned

experiment in previous works [6, 7] which give rise to a valley-selective optical

Stark effect.

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Chapter 6

Large, Valley-Exclusive Bloch-Siegert Shiftin Monolayer WS2

Coherent interaction with off-resonance light can be used to shift the energy levels

of atoms, molecules, and solids. The dominant effect is the optical Stark shift, but

there is an additional contribution from the so-called Bloch-Siegert shift that has

eluded direct and exclusive observation so far, particularly in solids. We observe an

exceptionally large Bloch-Siegert shift in monolayer WS2 under infrared optical

driving [1]. By controlling the light helicity, we can confine the Bloch-Siegert shift

to occur only at one valley and the optical Stark shift at the other valley, because the

two effects are found to obey opposite selection rules at different valleys. Such a

large and valley-exclusive Bloch-Siegert shift allows for enhanced control over the

valleytronic properties of two-dimensional materials.

6.1 Bloch-Siegert Shift

The fundamental interaction between light and matter can be understood within the

framework of a two-level system with an energy splitting E0 [2, 3]. Driving the

system by off-resonance light of frequency hω<E0 produces a series of photon-

dressed states (Floquet states) that are evenly spaced by hω, where h is the reduced

Planck’s constant. In the first order, there are two pairs of photon-dressed states – one



77

pair between the two original states (Fig. 6.1a) and the other pair outside the original

states (Fig. 6.1b). Interactions between the original states and these photon-dressed

states can be understood in terms of state repulsion. The former case leads to a shift of

transition energy called the optical Stark (OS) shift, which increases linearly with the

light intensity E20

� �and inversely with the detuning energy, ΔEOS / E2

0= E0 � hωð Þ[4]. The latter case also leads to a shift, called the Bloch-Siegert (BS) shift, which has

a different energy dependence,ΔEBS / E20= E0 þ hωð Þ [5]. Although the Bloch-

Siegert shift is negligible at small detuning, at large detuning it can become compa-

rable to the optical Stark shift.

The Bloch-Siegert shift has played an important role in atomic physics, notably

for its manifestation as the Lamb shift in quantum electrodynamics [6, 7] and its

Fig. 6.1 Comparison of the optical Stark shift and the Bloch-Siegert shift in a two-level system.

(a) Energy diagram for optical Stark (OS) shift. |ai and |bi denote the two original states with

resonance energy E0 before they are optically driven. |a+ hωi and |b� hωi are photon-dressed

(Floquet) states driven by the co-rotating optical field. Hybridization between these Floquet and

original states causes the resonance energy to blueshift by ΔEOS, which is proportional to the light

intensity E20

� �and inversely proportional to (E0� hω). (b) Energy diagram for Bloch-Siegert

(BS) shift. |a� hωi and |b + hωi are two different Floquet states driven by the counter-rotating

optical field. Hybridization between these Floquet and original states causes the Bloch-Siegert

shift, with magnitude ΔEBS inversely proportional to (E0� hω)

78 6 Large, Valley-Exclusive Bloch-Siegert Shift in Monolayer WS2

contribution to the trapping potential for cold atoms [8]. In condensed matter

physics, however, the Bloch-Siegert shift is a very rare finding because of its

typically small magnitude (<1 μeV); so far it has been revealed only indirectly in

artificial atoms by subtracting the dominant optical Stark shift using sophisticated

modeling [9–11]. To elucidate the detailed characteristic of Bloch-Siegert shift, it is

necessary to separate the two effects. Given that they are time-reversed partners of

each other – the optical Stark shift arises from co-rotating field and the Bloch-

Siegert shift from counter-rotating field – it is theoretically possible to separate

them under stimulation that breaks time-reversal symmetry.

6.1.1 Semiclassical Description

The energy shift that is induced by an off-resonance pump pulse can be obtained

from the time-dependent perturbation theory, from which the optical Stark

(OS) shift and the Bloch-Siegert (BS) shift usually appear together. In a small

detuning case, the OS shift dominates, and the BS shift only comes as a very small

correction term that is often neglected. In a large detuning case, however, the BS

shift can become comparable, and it must be taken into account. Here, we derive the

contribution of the BS shift at large detuning case, and by properly considering the

matrix elements, we show that the BS shift also follows a selection rule but with

helicity opposite to that of the OS shift. First, we will express the left-circularly

polarized light (pump) in terms of the co-rotating and counter-rotating fields.

Second, we consider the dipole approximation for the perturbation Hamiltonian

and obtain the respective matrix elements at the K and K0 valleys. Finally, we usethe time-dependent perturbation theory to obtain the respective energy shift at the K

and K0 valleys.A left-circularly polarized light can be expressed as

~ELCP ¼ E0 cos kz� ωtð Þx þ sin kz� ωtð Þyð Þ¼ E0

1

2eiωt þ e�iωt� �

x � 1

2ieiωt � e�iωt� �

y

� �

¼ E0

1

2

�x � iy

�e�iωt þ �

x þ iy�eiωt

� � ð6:1Þ

where E0 and ω are the electric field amplitude and the angular frequency of the

light. We have used (z ¼ 0) for monolayer material and used the Euler’s formula to

obtain the final expression. The perturbation Hamiltonian that is induced by ~ELCP

yields

Hab ¼ b e~r � ~ELCP

�� aD E¼ 1

2eE0 b

�~x� i~y

�e�iωt þ �

~xþ i~y�eiωt

�� a� � ð6:2Þ

The outcome of this perturbation depends on the particular states |ai and |bi. Inmonolayer WS2, there are two optical transitions we can consider, one at the K

6.1 Bloch-Siegert Shift 79

valley (Δm¼ � 1) and one at the K0 valley (Δm¼ + 1), and each of them obeys an

opposite selection rule with different helicities. At K valley, the non-zero matrix

element is given by only the first term in the bracket

Hab Kð Þ ¼ 1

2eE0 b;K

�~x� i~y

�� a;K� �e�iωt. At K0 valley, the non-zero matrix ele-

ment is given by only the second term in the bracket

Hab K0ð Þ ¼ 1

2eE0 b;K0 �~xþ i~y

�� a;K0� �eiωt:

Hab Kð Þ ¼ e�iωtμKE0=2 ð6:3ÞHab K0ð Þ ¼ eiωtμK0E0=2 ð6:4Þ

where μK ¼ e b;K�~x� i~y

�� a;K� �and μK0 ¼ e b;K0 �~xþ i~y

�� a;K0� �are the dipole

matrix elements at the two valleys, and they have equivalent magnitudes

μKj j ¼ μK0j j.By using the resulting perturbation Hamiltonians, we can evaluate the induced

energy shift of state |ai at the K valley through the standard time-dependent

perturbation theory [12]:

ΔEa tð Þ ¼ H∗ba tð Þe�iω0t

1

ih

ðt

0

Hab t0ð Þeiω0t0dt0

¼ eiωtμKE0=2ð Þe�iω0t1

ih

ðt

0

e�iωt0μ∗KE0=2

eiω0t0dt0

¼ 1

4μKj j2E0

2e�i ω0�ωð Þt 1ih

ðt

0

ei ω0�ωð Þt0dt0

¼ 1

4μKj j2E0

2e�i ω0�ωð Þt 1ih

ei ω0�ωð Þt � 1

i ω0 � ωð Þ� �

¼ � μKj j2E02

4h1� e�i ω0�ωð Þt

ω0 � ωð Þ� �

ð6:5Þ

where hω0(¼E0) is the energy gap. As it appears, the energy shift is time dependent

which oscillates at frequency (ω0�ω). In practice, the energy shift should saturate

at its mean value, which can be obtained from averaging through time T!1, as

follows:

ΔEa ¼ limT!1

1

T

ðT

0

ΔEa tð Þdt

¼ � μKj j2E02

4h ω0 � ωð Þ limT!1

1

TT � e�i ω0�ωð ÞT � 1

�i ω0 � ωð Þ� �� ð6:6Þ


where the second term in the square bracket is oscillating around finite values, but

this term eventually vanishes after it is divided by T!1. Similar expression can be

obtained for state |bi but with an opposite sign, ΔEb ¼ �ΔEa. Hence, the gap

between the two states at K valley becomes wider by an energy shift of

ΔEK ¼ μKj j2E02

4h1

ω0 � ωð6:7Þ

Similarly, we can also evaluate the induced energy shift at the K0 valley

ΔEa tð Þ ¼ H∗ba tð Þe�iω0t

1

ih

ðt

0

Hab t0ð Þeiω0t0dt0

¼ e�iωtμK0E0=2ð Þe�iω0t1

ih

ðt

0

eiωt0μ∗K0E0=2

eiω0t

0dt0

¼ 1

4μK0j j2E0

2e�i ω0þωð Þt 1ih

ðt

0

ei ω0þωð Þt0dt0

¼ 1

4μK0j j2E0

2e�i ω0þωð Þt 1ih

ei ω0þωð Þt � 1

i ω0 þ ωð Þ� �

¼ � μK0j j2E02

4h1� e�i ω0þωð Þt

ω0 þ ωð Þ� �

ð6:8Þ

The energy shift should also saturate at its mean value that can be obtained by a

similar procedure earlier. Finally, the gap between the two states at K0 valleybecomes wider by an energy shift of

ΔEK0 ¼ μK0j j2E02

4h1

ω0 þ ωð6:9Þ

Note that the energy shifts at both valleys have a similar expression, but with

different energy denominator, (ω0�ω) at K valley and (ω0 +ω) at K0 valley. Theformer energy shift corresponds to the ordinary optical Stark shift, while the latter

energy shift corresponds to the Bloch-Siegert shift. By virtue of the valley selection

rules, these two processes occur valley exclusively at opposite valleys when

induced by left-circularly polarized light (pump), that is, an ordinary optical Stark

shift at K valley and a Bloch-Siegert shift at K0 valley.


6.1.2 Quantum-Mechanical Description

The interaction between a two-level system and light can be described by the

interaction Hamiltonian Hint ¼ d � E ¼ 12hg σþ þ σ�ð Þ aþ a{

� �, where g is the cou-

pling constant. The total Hamiltonian yields

H ¼ 1

2hω0σz þ hωa{aþ 1

2hg σþaþ σ�a{ þ σþa{ þ σ�a

� � ð6:10Þ

where the first term corresponds to the two-level system and the second term to the

photon reservoir. In the interaction term, we have a combination of operators that

involves an excitation (σ+) or deexcitation (σ�) of the two-level system, and an

annihilation (a) or creation (a{) of photon. Among these, the first two terms

conserve energy, (σ+a + σ�a{), whereas the last two terms do not conserve energy,

(σ+a{ + σ�a). These are the two processes that give rise to the “ordinary” optical

Stark shift and the Bloch-Siegert shift of the energy resonance, respectively. In a

two-level system that obeys the Δm¼ 0 selection rule and interacts with linearly

polarized light, such as this one, the two processes cannot be separated. Thus, the

two contributions should be considered together to obtain an accurate shift of the

resonance, regardless of the choice of light polarization. At small detuning, the

non-energy conserving term is negligible and therefore neglected in the so-called

rotating wave approximation (RWA). This results in the original formulation of the

Jaynes-Cummings model (JCM) that can be solved exactly [13]:

HJCM ¼ 1

2hω0σz þ hωa{aþ 1

2hg σþaþ σ�a{

� � ð6:11Þ

Due to the interaction term, this Hamiltonian only couples states |a, n+ 1i and |b, ni,which we can use as the basis for the Hamiltonian matrix:

HJCM ¼ hω nþ 1

2

� �1 0

0 1

� �þ h2

ω0 � ω gffiffiffiffiffiffiffiffiffiffiffinþ 1

pg


p � ω0 � ωð Þ� �

ð6:12Þ

This Hamiltonian matrix can be diagonalized to give the energies:

En,� ¼ hω nþ 1

2

� �� h2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω0 � ωð Þ2 þ g2 nþ 1ð Þ

qð6:13Þ

As can be seen, the energy level exhibits a shift, called the optical Stark shift. The

shift of the |ai!|bi optical transition can also be evaluated, after taking a weak fieldapproximation in which g


p � ω0 � ω, giving

ΔEJCM ¼ h2

g2 nþ 1ð Þω0 � ω

ð6:14Þ


where (n+ 1) is proportional to the light intensity hε2i, which means that the energy

shift increases linearly with pump fluence at low-fluence regime. The shift is

inversely proportional to ω0�ω and becomes very effective at small detuning.

In a two-level system that obeys the Δm¼ � 1 selection rules (monolayer WS2),

however, the choice of light polarization must be considered because it has distinct

contributions [14]. This allows us to obtain exact solutions without the need to

consider the RWA. The more general Hamiltonian can be expressed in terms of the

canonical momentum ( p� eA), where A is the vector potential, and this yields an

interaction Hamiltonian:

Hint ¼ e=2mð Þ A � pþ p � Að Þ þ e2=2m� �

A2 ð6:15Þwhere we keep the first term and ignore the second (diamagnetic A2) term. The

momentum operator p can be written in terms of Pauli matrices (σ�) and evaluatedfor transition Δm¼ � 1 (at K or K0 valley) using the usual Cartesian-to-spherical

coordinates transformation, yielding

p�1 ¼ p�x þ iy

�σþ þ p∗

�x � iy

�σ� ð6:16Þ

pþ1 ¼ p�x � iy

�σþ þ p∗

�x þ iy

�σ� ð6:17Þ

where x and y are unit vectors, and p¼hb|px|ai is the dipole matrix element

between the ground state |ai and the excited state |bi. In our experiments, we

use pump light with left-handed circular polarization that has the form of

A � �x � iy

�a� þ �

x � iy�∗

a{�h i

, where a� and a{� are the annihilation and

creation operators for left-handed photon. Hence, we can evaluate the interaction

Hamiltonian at K (Δm¼ � 1) and K0(Δm¼ + 1) valleys separately:

Hint Kð Þ ¼ e=mð ÞA � p�1 ¼ hg σþa� þ σ�a{��

=2 ð6:18ÞHint K

0ð Þ ¼ e=mð ÞA � pþ1 ¼ hg σþa{� þ σ�a��

=2 ð6:19ÞThe total Hamiltonian at respective valleys can then be written:

H Kð Þ ¼ 1

2hω0σz þ hωa{�a� þ 1

2hg σþa� þ σ�a{�

� � ð6:20Þ

H K0ð Þ ¼ 1

2hω0σz þ hωa{�a� þ 1

2hg σþa{� þ σ�a

� � ð6:21Þ

where the energy-conserving interaction term now appears at K valley, while the

non-energy-conserving term appears at K0 valley. The Hamiltonian at K valley has

the same form with the exactly solvable JCM, and it gives rise to the “ordinary”

optical Stark shift of the energy resonance as shown earlier. Meanwhile, the

Hamiltonian at K0 valley can also be solved exactly. This Hamiltonian only couples


states |a, n� 1i and |b, ni, where n is the number of left-handed photons, and we can

use these two states as the basis for the Hamiltonian matrix:

HK0 ¼ hω n� 1

2

� �1 0

0 1

� �þ h2

ω0 þ ω gffiffiffin

pg

ffiffiffin

p � ω0 þ ωð Þ� �

ð6:22Þ

This Hamiltonian matrix can be diagonalized to obtain a set of dressed states with

energies:

En,� ¼ hω n� 1

2

� �� h2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω0 þ ωð Þ2 þ g2n

qð6:23Þ

At low pump fluence, the shift of the energy gap can be evaluated in a simpler form

after taking a weak field approximation, gffiffiffin

p � ω0 þ ω, yielding

ΔEK0 ¼ h2

g2n

ω0 þ ωð6:24Þ

where n is proportional to the light intensity hE2i. This is the Bloch-Siegert shift,

and there are three factors that govern the magnitude. First, the coupling constant

contains a dipole matrix element that requires an angular momentum change of

Δm¼ + 1. Second, the photon number n/hE2i means that the energy shift

increases linearly with pump fluence at low-fluence regime. Third, the shift is

inversely proportional to ω0 +ω and becomes more effective at pumping with

lower photon energies. This is the reason why this shift can be observed more

clearly using intense pump fluence at the infrared.


In our experiment, we investigate high-quality WS2 monolayers grown by chemical

vapor deposition (CVD) on sapphire substrates [15, 16]. We employ femtosecond

pump-probe absorption spectroscopy at ambient conditions. To reveal the Bloch-

Siegert shift, we pump the samples with strong infrared pulses at hω¼ 0.59 –

0.98 eV, far below the exciton resonance (E0¼ 2.0 eV). The induced exciton shift

is probed by the reflection of synchronized broadband pulses in the visible range

(1.83–2.17 eV), from which we can extract the absorption change. A blueshift of the

exciton absorption peak (α) is manifested as a differential curve in the absorption

change Δα (Fig. 6.2c). From this, we can deduce the magnitude of the energy shift.

In our experiment, we used pump fluence of up to 800 μJ/cm2 with pulse width of

160 fs. This gives the peak intensity of I ¼ fluence=pulsewidth ¼ E0cE20=2, from

which we can determine the peak intensity to be I¼ 5 GW/cm2 and the electric field

strength to be ε0 ¼ 2 MV/cm.


6.3 Observation of the Bloch-Siegert Shift

We report on the observation of a large Bloch-Siegert shift (ΔEBS ~ 10 meV),

which can be entirely separated from the optical Stark shift. Such a large and

exclusive Bloch-Siegert shift is realized in a monolayer of transition-metal

dichalcogenide (TMD) tungsten disulfide (WS2). This is possible because this

material system possesses two distinctive features. First, it exhibits strong light-

exciton interaction at the two time-reversed valleys (K, K0) in the Brillouin zone

(Fig. 6.2a–c) [17–19]. Secondly, the two valleys possess finite and opposite Berry

curvature owing to the lack of inversion symmetry, giving rise to distinct optical

selection rules and related valleytronic properties [20–28]. That is, the optical

transition at the K (K0) valley is coupled exclusively to left-handed σ� (right-

handed σ+) circularly polarized light. Such a unique material platform allows us

to separate the Bloch-Siegert shift from the optical Stark shift by using circularly

polarized light.

We employ femtosecond pump-probe absorption spectroscopy in our experi-

ment (Fig. 6.2a). All measurements were carried out at room temperature. We pump

Fig. 6.2 Observation of valley-exclusive Bloch-Siegert shift in monolayer WS2. (a) Illustration ofthe pump-probe experiment. We pump the monolayer WS2 sample with strong infrared pulses and

measure the pump-induced change of reflection with broadband probe pulses. (b) The K and K0

valleys in monolayer WS2. Optical pumping with left-handed circular polarization (σ�) couplesonly to the K valley, and not the K0 valley, unless the counter-rotating field is taken into account.

(c) Measured A exciton absorption spectrum (α, black curve in the top panel) of monolayer WS2 in

equilibrium. The dashed curve represents the shifted resonance (simulated) under red-detuned

optical pumping. This shift produces a differential curve in the absorption change (Δα, red curve inthe bottom panel)

6.3 Observation of the Bloch-Siegert Shift 85

a monolayer of WS2 with intense infrared light pulses and probe the energy shift at

the K (K0) valley with σ�( σ+) visible light pulses. A blueshift of the exciton

absorption peak (α) is manifested as a differential curve in the absorption change

(Fig. 6.2c). From this, we can deduce the magnitude of the energy shift at both

valleys. Previously, transient absorption with visible pumping has been used to

study the optical Stark effect in monolayer TMDs [29, 30]. Here, by pumping with

infrared light, we reveal the Bloch-Siegert shift in WS2.

Figure 6.3a–d displays our results at the K valley (blue curve) and K0 valley (redcurve). For comparison, we first show the Δα spectra under zero-delay pumping at

hω ¼ 1.82 eV (Fig. 6.3a). For this small detuning energy, only the K valley shows

an appreciable signal. This signal arises from the optical Stark shift, which occurs

exclusively at the K valley [29, 30]. The K0 valley exhibits only very weak (but

observable) signal. However, as we lower the pumping photon energy to 0.59 eV,

the signal at the K0 valley becomes comparable to the signal at the K valley

Fig. 6.3 (a, b) The Δα spectra under zero-delay optical pumping at pump photon energy 1.82 and

0.59 eV. By using probe pulses with σ� (σ+) polarization, we can selectively measure Δα at the K

(K0) valley, as shown in the left (right) column. The black curves in (E) are smoothened curves to

average out the modulations, see text. (c) Time trace ofΔα spectra in (E) measured at probe energy

of 1.96 eV and pump energy of 0.59 eV. The induced energy shift is observed only at zero time

delay. (d) The zero-delay Δα spectra of the K valley (blue curves) and K0 valley (red curves) underdifferent incident pump fluence (ℱ¼ 160 – 800 μJ/cm2). The pump photon energy is 0.59 eV. The

spectra are vertically displaced for clarity


(Fig. 6.3b). This observation indicates a pronounced energy blueshift at the K0

valley, a phenomenon that apparently violates the well-established valley selection

rules in monolayer TMDs. Some modulations also appear in the Δα spectrum.

These minor features are possibly induced by electron-phonon coupling and war-

rant further investigation; here we average out these modulations by smoothening

the curves (black lines). We have further examined the signals at different pump-

probe time delays. The Δα signals at both valleys emerge only at zero time delay,

with very similar temporal profiles to the 160 fs duration of the pump pulses

(Fig. 6.3c). These results indicate the coherent nature of the energy shift and also

exclude the influence of intervalley scattering from possible excited carriers that

typically occurs on the picosecond timescale [31–33].

6.4 Fluence and Detuning Dependences

To investigate the underlying mechanism of the anomalous energy shift at K0

valley, we have measured the zero-delay Δα spectra for both valleys at various

pump photon energies (hω¼ 0.59, 0.69, 0.89, 0.98 eV) and different pump fluences

(ℱ¼ 30 – 800 μJ/cm2). Here we display the fluence-dependent spectra for pump

photon energy hω¼ 0.59 eV (Fig. 6.3d). The spectra at both valleys are found to

grow with increasing pump fluence. For a more quantitative analysis, we have

extracted the energy shift from each spectrum and plotted it as a function of

ℱ/(E0� hω) (Fig. 6.4a). The shift at K valley exhibits an excellent linear depen-

dence regardless of the different pump photon energies (closed symbols), indicating

that it arises from the optical Stark effect. The shift at K0 valley, however, spreadsout with no rigorous linear dependence (open symbols). Such a contrasting behav-

ior indicates that the K0-valley shift does not arise from the optical Stark effect. In

Fig. 6.4b, we replot the K0-valley shift as a function of ℱ/(E0 + hω) with the same

axes scales; the data now exhibit an excellent linear dependence. Moreover, the

slope of the K0-valley shift in this new plot is identical to the slope of the K-valley

shift in (Fig. 6.4a). This observation strongly suggests that the K0-valley shift arisesfrom the Bloch-Siegert effect.

6.5 Valley-Exclusive Optical Stark Shift and Bloch-SiegertShift

Our finding can be verified quantitatively by using either a semiclassical theory or a

fully quantum-mechanical theory [14] (Sect. 6.1). As we probe only the lowest-

energy exciton state (1 s), which shows similar properties as those of hydrogen

atoms, it is appropriate and sufficient to use a simple two-level framework, as

shown in earlier studies [18, 21, 29, 30]. In our semiclassical analysis, we treat the

6.5 Valley-Exclusive Optical Stark Shift and Bloch-Siegert Shift 87

ground state and the 1 s exciton state as the two-level system (|ai and |bi) with a

resonance energy, driven by a classical electromagnetic wave with amplitude ε0 andfrequency ω. We use a left-circularly polarized pump beam~E tð Þ ¼ E0 cos kz� ωtð Þx þ sin kz� ωtð Þyð Þ, polarized along the xy-plane of the

monolayer sample (z ¼ 0), which can also be expressed as

~E tð Þ ¼ 1

2E0

�x � iy

�e�iωt þ �

x þ iy�eiωt

� � ð6:25Þ

where the field is decomposed into two terms based on their time evolution. The

interaction Hamiltonian can then be expressed as

Hab ¼ b e~E �~r�� aD E

¼ 1

2eE0 b

�~x� i~y

�e�iωt þ �

~xþ i~y�eiωt

�� a� �� ð6:26Þ

Here the first term�~x� i~y

�e�iωt induces a transition with Δm¼ � 1 (co-rotating

field), and the second term�~xþ i~y

�eiωt induces a transition with Δm¼ + 1

(counter-rotating field). Owing to the unique valley selection rules in

monolayer WS2, these two terms are thus coupled exclusively to the K(Δm¼ � 1)

Fig. 6.4 Fluence and detuning dependences of the Bloch-Siegert shift. (a) Energy shifts at the K

and K0 valleys (closed and open symbols) as a function of Fluence/(E0� hω). The data are

extracted from the spectra in Fig. 6.3. The K-valley shift exhibits a rigorous linear dependence

(solid blue line), but the K0-valley shift spreads out (red region). (b) Energy shift at K0 valley as in(A), but plotted as a function of Fluence/(E0 + hω). The linear dependence becomes obvious. The

top-left inset shows the predicted ratioΔEBS/ΔEOS (blue line) when the detuning energy (E0� hω)increases from zero to the resonant energy (E0 ¼ 2 eV for monolayer WS2). The open circles are

our averaged experimental data obtained from Fig. 6.3


and K0(Δm¼ + 1) valleys, respectively (Fig. 6.5a, b), with their valley-specific

interactions as

Hab Kð Þ ¼ e�iωtμKE0=2 ð6:27Þ

Hab K0

¼ eiωtμK0E0=2 ð6:28Þ

Here μK and μK0 are the dipole matrix elements at the K and K0 valley, respectively,and they have equal magnitudes μ ¼ μKj j ¼ μK0j j. However, they are associated

with opposite time-evolution factors, which lead to a more general theory of valley

selection rules in monolayer TMDs. Under resonant absorption condition

(hω¼E0), the left-circularly polarized light couples only to the K-valley. But

under off-resonance condition (hω<E0), the coupling to the K0-valley can become

appreciable through the time-reversed process, giving rise to noticeable energy

shift. The induced energy shifts at the respective valleys can be evaluated by the

time-dependent perturbation theory as

ΔEK ¼ μ2E20

2

1

E0 � hωð6:29Þ

ΔEK0 ¼ μ2E20

2

1

E0 þ hωð6:30Þ

The two energy shifts have different energy dependence, from which we can readily

identify ΔEK to be the optical Stark shift and ΔEK0 the Bloch-Siegert shift. When

plotted as a function of their respective energy denominator (E0� hω, or E0 + hω),

Fig. 6.5 (a) Energy diagram before and after optical pumping for the optical Stark shift, which

occurs only at the K valley. (b) Energy diagram for the Bloch-Siegert shift, which occurs only at

the K0 valley

6.5 Valley-Exclusive Optical Stark Shift and Bloch-Siegert Shift 89

both shifts exhibit an identical slope. The prediction of common slope and opposite

valley indices agrees well with our experimental observation (Fig. 6.4a–b). From

our data, we can deduce the dipole matrix elements to be μ¼ 55 Debye, in excellent

agreement with previous measurements [29]. In addition, the ratio between ΔEK0

and ΔEK is predicted to be (E0� hω)/(E0 + hω), the same as ΔEBS/ΔEOS for a

two-level system with no valley degree of freedom that we discussed earlier. By

plotting the average shift ratio measured for each pump photon energy, we find a

good agreement between our experiment and theory (Fig. 6.4b inset).

The physics of this valley-exclusive energy shift can be illustrated in the energy

diagrams shown in Fig. 6.5a, b. The co-rotating field generates a Floquet state hωabove the ground state in both valleys, with energy separation E0� hω from the

excited state. Due to the matching condition of angular momentum, repulsion

between the Floquet state and the excited state only occurs at the K valley, giving

rise to the ordinary optical Stark shift (Fig. 6.5a). On the other hand, the counter-

rotating field generates a Floquet state below the ground state, with energy separa-

tion E0+ hω from the excited state (Fig. 6.5b). The matching condition of angular

momentum forbids the level repulsion at the K valley but allows it at the K0 valley.This gives rise to the Bloch-Siegert shift at the opposite (K0) valley. In other words,the left-circularly polarized light can be understood as stimulating the σ� absorp-

tion (Δm¼ � 1) and σ� emission (Δm¼ + 1) processes at the K and K0 valleys,respectively. This makes it possible for the circularly polarized light with a given

helicity to couple to both valleys in a distinct manner, thus enriching the valley

selection rules.

6.6 Conclusions

The Bloch-Siegert shift we observed exhibits the opposite valley selection rules

from the ordinary optical Stark effect, which allows us to completely separate the

two effects. This is possible because, as time-reversed partners, the two effects

share similar relationship with the two time-reversed valleys in monolayer WS2,

which can be disentangled under circularly polarized light that breaks the time-

reversal symmetry. Our finding reveals more general valley selection rules and

promises enhanced control over the valleytronic properties of Dirac materials, such

as graphene, TMDs, as well as Weyl semimetals. Furthermore, by using higher

pump intensity or lower pump photon energy with selective light helicity, we

expect to reveal higher-order coherent effects that could produce polarized valley

current in transport experiments [34].


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ultrastrong coupling regime. Phys. Rev. Lett. 105, 237001 (2010)

11. J. Tuorila et al., Stark effect and generalized Bloch-Siegert shift in a strongly driven two-level

system. Phys. Rev. Lett. 105, 257003 (2010)

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Chapter 7

Lennard-Jones-Like Potential of 2D Excitonsin Monolayer WS2

In this chapter, we report a rare atom-like interaction between excitons in mono-

layer WS2, measured using ultrafast absorption spectroscopy. At increasing exci-

tation density, the exciton resonance energy exhibits a pronounced redshift

followed by an anomalous blueshift. Using both material-realistic computation

and phenomenological modeling, we attribute this observation to plasma effects

and an attraction-repulsion crossover of the exciton-exciton interaction that mimics

the Lennard-Jones potential between atoms. Our experiment demonstrates a strong

analogy between excitons and atoms with respect to inter-particle interaction,

which holds promise to pursue the predicted liquid and crystalline phases of

excitons in two-dimensional materials.



93

7.1 Many-Body Interactions in 2D TMDs

Excitons in semiconductors are often perceived as the solid-state analogs to hydro-

gen atoms. This analogy helps us to understand the basic features of excitons,

notably their internal energy states. However, this analogy breaks down as we

consider the inter-particle interactions because of some fundamental differences

between atoms and excitons. Atoms are stable particles with large ionization energy

(~10 eV). They exhibit long-range van der Waals attraction and short-range Pauli

repulsion, which form the so-called Lennard-Jones potential as a function of

interatomic separation [1, 2]. In contrast, excitons are transient quasiparticles

with much smaller binding energy and extremely short lifetime. They can dissoci-

ate into an electron-hole plasma, whose relative concentration is governed by the

law of mass action [3, 4]. Hence, interactions in semiconductors are somewhat

different from those in real gases, because the effects from plasma [5] and excitons

[6] must be considered. The relative importance of exciton and plasma effects

depends on the regime of excitation density. In conventional III-V and II-VI semi-

conductors, such complex many-body effects preclude the demonstration of atom-

like interactions between excitons, particularly in the regime of high excitation

density where excitons become unstable.

Recent advances in two-dimensional (2D) semiconductors, particularly mono-

layer transition metal dichalcogenides (TMDs), offer a unique platform to investi-

gate excitonic interactions. These materials possess strong Coulomb interactions

due to quantum confinement and reduced dielectric screening [5–7], leading to the

formation of excitons with exceptionally large binding energies ~300 meV [6–

11]. The enhanced stability of excitons in these materials provides good opportu-

nities to reexamine the role of plasma effects and excitonic interactions over a broad

range of excitation density. Although the exciton physics in photoexcited TMDs

has been much studied [10–17], a complete picture of excitonic interactions in these

materials is still lacking.

Here we investigate systematically the many-particle interactions in monolayer

WS2. We combine ultrafast absorption spectroscopy, microscopic many-body

theory, and an analytic approach that maps the measured exciton-exciton interac-

tions onto an effective atomic model. In particular, we measure the absorption

spectrum of the A exciton under femtosecond optical excitation. As we increase the

excitation density, we observe a pronounced redshift (73 meV) of the exciton

resonance energy at low density, followed by an unusual blueshift (10 meV) at

high density. We attribute the two different energy shifts to two distinct interaction

regimes. At high density, the exciton blueshift is well described by assuming a

repulsive exciton-exciton interaction similar to the short-range Lennard-Jones

interaction between atoms [1, 2]. At low density, the Lennard-Jones potential

further contains a long-range contribution due to an attractive exciton-exciton

interaction. However, in contrast to the atomic case, the exciton redshift in this

regime is found to follow a strongly modified exponent, indicating that not all

carriers are bound excitons but a fraction exists as electron-hole plasma. Insight

94 7 Lennard-Jones-Like Potential of 2D Excitons in Monolayer WS2

from microscopic theory reveals that the redshift observed at low excitation density

mainly arises from plasma-induced bandgap renormalization and screening of the

exciton binding energy. We note that the observed energy shift is much larger than

those reported in conventional semiconductors such as GaAs quantum wells

(~0.1 meV) [16–22] as a consequence of the much enhanced many-body interac-

tions in the 2D material. Our model is further supported by the temperature-

dependent exciton energy shift observed in the time-resolved absorption

measurements.


The sample consists of high-quality monolayers of WS2 that were grown on

sapphire substrates by chemical vapor deposition (CVD) [23, 24]. All measure-

ments in this study were conducted at ambient condition (300 K, 1 atm). In our

experiments, we used a Ti/sapphire amplifier producing laser pulses with duration

of 50 fs and at 30 kHz repetition rate. Each pulse was split into two arms. For the

pump arm, the pulses were sent to a second-harmonic nonlinear crystal, while for

the probe arm the pulses were sent through a delay stage and a white light

continuum generator (hv ¼ 1.78–2.48 eV, chirp-corrected). The two beams were

focused at the sample, and the probe beam was reflected to a monochromator and a

photodiode for lock-in detection [14, 25]. By scanning the grating and the delay

stage, we were able to measure ΔR/R (and hence α [14]) as a function of energy and

time delay.

In our pump-probe experiment, we generate carriers by femtosecond pump

pulses with photon energy hv ¼ 3.16 eV, which is above the quasiparticle bandgap

at around 2.3 eV at room temperature [11]. The resulting A exciton resonance near

EA ¼ 2.0 eV is monitored through the reflection spectrum of broadband probe

pulses with controlled time delay at room temperature (Fig. 7.1a). For a monolayer

sample on a transparent substrate, the absorption spectrum can be extracted from

the reflection spectrum using thin-film approximation (see Chap. 2, Sect. 2.2). We

estimate the excitation density n from the measured incident pump fluence and

absorbance of the sample at the excitation wavelength [24–26].

7.3 Optical Signature of Many-Body Effects

Figure 7.1b shows the absorption spectra of the A exciton at increasing pump

fluence up to 18 μJ/cm2 (n ¼ 5.3� 1012 cm�2). The spectra were taken at a

pump-probe delay of 2 ps, a time after which the excitons are expected to have

reached thermal equilibrium with one another and with the lattice but not yet

recombined [27, 28]. All the spectra can be fitted well with a Lorentzian function

7.3 Optical Signature of Many-Body Effects 95

plus a second-order polynomial function (smooth lines over the data curves), which

represent the exciton peak and the background slope, respectively:

α ωð Þ ¼ I0g0

π ω� ω0ð Þ2 þ g20

� �0@ 1Aþ Aþ B ω� ω1ð Þ þ C ω� ω1ð Þ2

� �ð7:1Þ

Here, I0, ω0, and g0 are the absorption peak intensity, energy, and linewidth,

respectively, while A, B, C, and ω1 are the background constants and energy

reference, respectively. We first use this expression to fit the equilibrium absorption

spectrum and record the fitting parameters. The obtained background parameters

are fixed for subsequent fitting procedures in fluence-dependent and time-

dependent spectra, and only the Lorentzian function parameters (I0, ω0, and g0)are allowed to vary. We note that the B exciton is well separated from the A exciton

by 400 meV in monolayer WS2 [26] and therefore does not affect our analysis.

From the fitting, we extract the exciton peak energy (EA), linewidth (Γ), peakintensity (I), and spectral weight (S, i.e., the integrated area) and plot their changes

in Fig. 7.2a–c. While the spectral weight remains unchanged at all excitation

densities, the other quantities vary significantly with the density.

7.3.1 Exciton Redshift-Blueshift Crossover

These quantities exhibit two distinct behaviors at low and high excitation densities.

At low density, the peak energy redshifts gradually for ~70 meV as the density

increases, while the linewidth and peak intensity remain almost constant. The

energy shift cannot be explained using the pump-induced lattice heating because

Fig. 7.1 (a) Schematic of transient absorption spectroscopy setup. (b) Absorption peak of A

exciton in monolayer WS2 at increasing excitation densities


the estimated lattice temperature increase is ~20 K, which corresponds to merely

~4 meV of temperature-dependent gap narrowing [29]. In contrast, with further

increase of density n> 2.0� 1012 cm�2, the rate of exciton redshift diminishes and

eventually turns into a blueshift of ~10 meV from the lowest energy point. Notably,

the redshift-blueshift crossover is accompanied by a large spectral broadening, with

the linewidth increasing to more than twice of the initial width (Fig. 7.2b). Corre-

spondingly, the peak intensity drops to one half of the initial intensity to maintain

the total spectral weight of the A exciton (Fig. 7.2c). These two distinct energy

shifts correspond to two different interaction regimes as we will discuss in the

following.

Fig. 7.2 (a) Exciton energyshift (ΔE) as a function of

pump fluence obtained from

the experiment (red) andcalculation (purple). (b)Linewidth broadening (ΔΓ/Γ0), (c) peak height (I/I0),and spectral weight change

(ΔS/S0)

7.3 Optical Signature of Many-Body Effects 97

7.3.2 At Low Density: Plasma Contribution

We first discuss the redshift at low density. According to prior studies in monolayer

TMDs [5], the exciton redshift can be ascribed to a combination of bandgap

renormalization and plasma screening of the exciton binding energy due to the

excited unbound carriers (Fig. 7.3). We have obtained the quasiparticle band

structure and Coulomb matrix elements of monolayer WS2 by first-principle

G0W0 calculations. Screening from the substrate is additionally incorporated in

the Coulomb matrix elements [28–30]. The exciton shift due to the excited carriers

is calculated with microscopic semiconductor Bloch equations in a screened-

exchange Coulomb-hole approximation (SXCH) – see Sect. 7.7.1. With increasing

excitation density, both the quasiparticle bandgap (Eg) and the exciton binding

energy (Eb) are found to decrease (Fig. 7.4a). Since Eg decreases faster than Eb, the

resulting exciton resonance energy (EA¼Eg�Eb) shifts to lower energies

(Fig. 7.4b).

Our calculations reproduce the measured redshift at low density (Fig. 7.2a,

purple curve). The overall agreement is remarkable, given that we do not use any

fitting parameter in our theory. We note that the SXCH calculation predicts a Mott

transition at n ¼ 2� 1012 cm�2. Approximately at this density, the experimental

data reveals a crossover into an anomalous blueshift (Fig. 7.2a), which we attribute

to exciton-exciton interaction that is facilitated by the increasing fraction of carriers

bound into excitons and which marks the limit in carrier density to which a plasma

picture applies. Although the observed redshift at low density is dominated by

plasma contribution, in a purely excitonic picture, a redshift can also result from

exciton-exciton attraction similar to atoms. Mutual attraction can reduce the energy

cost to create an extra exciton (EA) by a magnitude as much as the negative inter-

exciton potential energy.

Fig. 7.3 Schematic band

diagram showing that the

ΔE results from an

incomplete compensation

between the gap narrowing

and the exciton binding

energy reduction


7.3.3 At High Density: Exciton Contribution

In light of the purely excitonic picture, we interpret the blueshift at high density

>8 uJ/cm2 (n > 2.0� 1012 cm�2), where a large fraction of carriers form bound

excitons, as arising from an exciton-exciton repulsion. Indeed, the blueshift is notcaptured by our numerical approach and therefore suggests a new contribution

other than the plasma effects. The simultaneous broadening of the absorption peak

indicates that this new interaction strongly perturbs the excitons and shortens their

lifetime (Fig. 7.2b). In this scenario, high-density excitons tend to repel each other

due to the Pauli exclusion of overlapping electron orbitals, giving rise to positive

inter-exciton potential energy. As a consequence, the energy cost to create an extra

exciton increases, leading to a blueshift of the resonance energy.

7.4 Lennard-Jones-Like Potential as an Effective Model

The above interpretation has inspired us to quantify the excitonic contributions to

the energy shift in the entire density range through a simple phenomenological

model with two power laws in analogy to the well-known Lennard-Jones potential

between atoms (for which k ¼ 6):

ΔE ¼ εr0rs

� �8

� r0rs

� �k" #

ð7:2Þ

Fig. 7.4 (a) Theoretical results on the gap narrowing (ΔEg) and the exciton binding energy (Eb) at

increasing exciton density (logscale). (b) Resulting exciton shift ΔE at increasing density

(logscale)

7.4 Lennard-Jones-Like Potential as an Effective Model 99

Here rs is the radius of disk occupied by an exciton nπr2s ¼ 1� �

, while ε, r0, and k arethe fitting parameters, which can be interpreted in a similar way as in the usual

“12–6” power-law potential between atoms. The first term describes the exciton

blueshift caused by the short-range Pauli repulsion. We use the r�8s functional form

for better fitting of the Pauli repulsion in this system instead of the usual r�12s

typically chosen in atomic system for convenience due to the relative computing

efficiency (r�12s is the square of r�6

s ). The second term models the exciton redshift

caused by the long-range van der Waals attraction of excitons behaving as fluctu-

ating dipoles in the presence of plasma in this material. In general, the functional

form of this attraction potential can differ from the usual London dispersion force

r�6s ; hence we parameterize it as r�k

s . By fitting the ΔE–rs data through the least

squares method with ε ¼ 128� 10 meV, r0 ¼ 2.6� 0.1 nm, and k ¼ 1.4� 0.2, our

simple model matches precisely the density dependence of exciton energy shift

(Fig. 7.5a). Note that exciton-exciton annihilation can be present in highly excited

monolayer TMDs [25–30, 33, 34]. The x-axis in Fig. 7.5a shows the rs from

estimated density that is already corrected after considering the annihilation effect

at 2 ps (Sect. 7.7.2). We note that the exponent k differs (down to one fifth) from the

exponent in the Lennard-Jones potential due to the presence of plasma effects in

semiconductors, which are not captured by the atomic model. The obtained

r0 ¼ 2.6 nm represents the exciton Bohr radius in monolayer WS2. This value

agrees well with our calculated exciton radius (2.3 nm) [11–13] and the estimated

radius (1–3 nm) in other studies [9–11, 33–36].

Excitons have been perceived as the solid-state counterpart of atoms, but the

analogy is usually drawn only for their similar internal structure and molecular

structures. The latter is apparent from the formation of trions [35–39] and

biexcitons [12–15, 38–41] with binding energies 20–60 meV in monolayer TMDs

that are analogous to the hydrogen anion and hydrogen molecules. Here, the good

agreement between the modified Lennard-Jones model and our experiments reveals

further that they also share similar mutual interaction behavior at long and short

distances. This finding is remarkable because high-density excitons are usually

unstable against the electron-hole plasma formation and other annihilation pro-

cesses. These competing processes can easily destroy the exciton resonance fea-

tures and hinder the observation of inter-exciton repulsion. Monolayer WS2 is,

however, an exceptional material, which hosts tightly bound excitons with radius

approaching the atomic limit. The robustness of these excitons helps maintain their

resonance features even at very high density. We can therefore observe an effective

attraction-repulsion crossover of excitonic interactions, a phenomenon that was

predicted early [40–42] but remained unobserved experimentally until now.

Although the observed shift mimics the Lennard-Jones potential, there are three

features distinct from atoms that deserve more careful attention. First, in addition to

excitons, plasma can be present simultaneously with a relative density governed by

the law of mass action. The inter-particle separation rs is derived from their

combined densities. Second, plasma contribution to the shift at low density dom-

inates the exciton contribution. The observed �1/rs dependence, instead of 1=r6s ,indicates a negligible contribution from exciton van der Waals attraction. This is


not surprising because excitons in monolayer TMDs are tightly bound. Third, a

possible formation of biexcitons at short distance is not explicitly captured in this

model. This is a similar situation faced by Lennard-Jones potential because it does

not explicitly represent chemical bonding between atoms, but it can explain why a

cluster of atoms can form at the minimum potential. Although biexciton formation

may occur at the minimum potential (Fig. 7.5a), the strong plasma screening

precludes such occurrence. This is evidenced from the much reduced exciton

binding energy at such a high excitation density (Fig. 7.4a). This means biexciton

binding energy should also be reduced to a value much smaller than the reported

values, or completely screened, and thus unlikely to form.

7.5 Chronological Signature of Interactionsin Time-Resolved Spectra

We can further test our model (Eq. 7.2) through its temperature-dependent behav-

ior. Two effects arise when the exciton temperature is high. First, such highly

energetic excitons will be in constant motion and dynamically average out their

Fig. 7.5 (a) Exciton energy shift (ΔE) as a function of average radius (rs) occupied by an exciton

in the exciton gas. The red dots are experimental data from Fig. 7.2a. The solid black line is thebest fit of our phenomenological model (Eq. 7.2). The dashed lines are the repulsion and attractioncomponents of the inter-exciton potential. r0 ¼ 2.6 nm is the extracted exciton radius. (b)Schematic configuration of a probe exciton (red) among the pump-generated excitons (purple)at different interaction regimes [1, 2, 31, 32] as denoted in panel (a)

7.5 Chronological Signature of Interactions in Time-Resolved Spectra 101

short-range and long-range interactions among each other. Secondly, the excitons

will have higher probability to reach their internal excited states. These effects will

reduce the effective potential energy and increase their Bohr radius. As a conse-

quence, the energy potential well between the excitons will become shallower, andthe inter-exciton distance at the potential minimum will become wider (inset of

Fig. 7.7a). In other words, when the excitons cool from very high to low temper-

ature, we predict a significant redshift of their energy. As we discuss below, our

model captures the complex cooling dynamics and offers a simple interpretation

based on such exciton picture, although rigorous contribution from plasma effects

could be included for a more accurate model.

This prediction can be conveniently explored in the cooling dynamics of exci-

tons after pump excitation at high density regime, where exciton-exciton interaction

dominates and the exciton picture is particularly appropriate. As we pump mono-

layer WS2 using 3.16 eV photons (hν>EA), we create free electron-hole pairs that

immediately form excitons within the excitation pulse duration [43, 44]. This is due

to the strong Coulomb attraction in monolayer WS2 that leads to a very rapid

exciton formation. The large excess energy will bring the excitons to a high

temperature (Te > 1000 K). Subsequently, we can follow the time evolution of

the exciton resonance as they cool down. Figure 7.6a, b shows a time series of the

exciton absorption spectra up to 1 ns (F ¼ 11 uJ/cm2), from which we extract their

peak parameters (Fig. 7.7a, b). By examining these parameters, the exciton dynam-

ics can be described roughly in three stages.

In Stage I (red region), the hot excitons are formed with rapidly decreasing inter-

exciton distance, accompanied by a dramatic energy redshift and spectral broaden-

ing. In Stage II (blue), the hot excitons cool to the lattice temperature via phonon

Fig. 7.6 Exciton absorption peak at increasing time delays from�0.30 to 0.48 ps (a) and from 0.5

to 1000 ps (b), where only the subset of the data is shown for clarity


emission. In Stage III (yellow), the excitons recombine gradually with increasing

inter-exciton distance leading to a blueshift. In the framework of our model, such

exciton dynamics can be adequately described by the trajectory (I!II!III)

between the hot and cold potential energy curves, as illustrated in the inset of

Fig. 7.7a. We exclude possible contribution from lattice cooling in Stage III

because the estimated temperature-dependent energy shift is merely ~4 meV

(20 K), far too small than the observed shift ~70 meV (Sect. 7.7.3). It is particularly

noteworthy to examine the dynamics in Stage II, where the excitons cool down

from Te> 1000 K to ~300 K, but the density should remain unchanged. As depicted

by arrow II in the inset of Fig. 7.7a, the exciton energy decreases when the high-

temperature potential curve (red) evolves into the low-temperature curve (blue).

During this process, the exciton energy is found to redshift for about 20 meV,

accompanied by a decrease of linewidth. With this interpretation, we can assign to

Stage II an exciton cooling time of 2 ps, comparable with the cooling time measured

in graphene [45].

Fig. 7.7 (a) Exciton energy shift (ΔE), as well as (b) peak height (I/I0), linewidth broadening (ΔΓ/Γ0), and spectral weight change (ΔS/S0). Inset in (a) shows the stages of relaxation dynamics

through the ΔE vs rs curve at high and low temperatures

7.5 Chronological Signature of Interactions in Time-Resolved Spectra 103

7.6 Summary

In summary, we have observed a transition of inter-excitonic interaction at increas-

ing exciton density in monolayer WS2, which manifests as a redshift-blueshift

crossover of the exciton resonance energy. At low density, the exciton redshift

arises from plasma screening effects and the long-range exciton-exciton attraction.

At high density, the exciton blueshift is attributed to the short-range exciton-exciton

repulsion. We describe this density dependence of the excitonic interactions by a

phenomenological model, in analogy to the Lennard-Jones interaction between

atoms, combined with a material-realistic computation of plasma effects.

Interpreting the time dependence of energy shifts shortly after the carrier excitation

in terms of our model, we extract an exciton cooling time of about 2 ps. Similar

results are also observed in monolayer MoS2, implying that this behavior is

ubiquitous in monolayer TMD semiconductors [12–14]. The close analogy between

the excitons and atoms, as shown in our experiment, suggests that the liquid and

crystal phases of excitons [44–48] can be realized in 2D materials.

7.7 Supplementary

7.7.1 Microscopic Many-Body Computation

To calculate linear optical properties of monolayer WS2 on a substrate under the

influence of excited carriers, we combine first-principle G0W0 calculations with the

solution of the semiconductor Bloch equations in screened-exchange Coulomb-

hole (SXCH) approximation for the two highest valence bands and the two lowest

conduction bands as described in detail and applied to freestanding monolayer

MoS2 in Ref. [3–5]. In the following, we describe in detail how the previously

used theory has to be augmented to properly take the substrate into account. We

assume that the substrate mainly affects the internal Coulomb interaction and

neglect its influence on the band structure, as we are only interested in relative

shifts of the exciton resonance energy. Therefore, we derive the bare Uαβ(q) andscreened Vαβ(q) Coulomb interaction matrices in the Wannier-orbital basis (with

α, β2 dz2 ; dxy; dx2�y2�

) for a freestanding WS2 slab using the FLEUR and SPEX

codes [29, 49]. As discussed in [30], macroscopic screening effects (like those

arising from substrates) are described by the leading or macroscopic eigenvalue ofthe dielectric matrix. To access this quantity, we transform the full matrices Uαβ(q)and Vαβ(q) to their diagonal representations Ud(q)¼TU(q)T∗ and Vd(q)¼TV(q)T∗

using the eigenbasis T of the bare interaction and define the diagonal dielectric

function via εd(q)¼Ud(q)/Vd(q). Now each diagonal matrix is defined by its three

eigenvalues. We fit the leading eigenvalues U1(q) and ε1(q) via


U1 qð Þ ¼ e2

2ε0A

1

q 1þ γqþ δq2ð Þ ð7:3Þ

ε1 qð Þ ¼ ε1 qð Þ 1� β1β2e�2qh

1þ β1 þ β2ð Þe�qh þ β1β2e�2qhð7:4Þ

while all other elements (U2,U3, ε2 and ε3) are approximated by constant values

given in Table 7.1.

In Eq. 7.3, e is the elementary charge, ε0 the vacuum permittivity, A the unit cell

area per orbital, and γ and δ are used to obtain optimal fits to the vacuum

extrapolated ab initio data. In Eq. 7.3, we introduced the parameters βi which are

given by

βi ¼ε1 qð Þ � εsub:, iε1 qð Þ þ εsub:, i

ð7:5Þ

Here, the dielectric constants of the substrate (i ¼ 1) and superstrate (i ¼ 2) are

introduced. In order to describe the original ab initio data as close as possible, we fit

ε1(q) using

ε1 qð Þ ¼ aþ q2

a sin qcqbc

þ q2þ e ð7:6Þ

and set εsub . , 1¼ εsub . , 2¼ 1. As soon as all fitting parameters are obtained (see

Table 7.1), the screening of a dielectric environment can be included by choosing

εsub . , 1 or εsub . , 2 correspondingly. In this paper, we use εsub . , 1¼ 10 (εsub . , 2¼ 1)

which models the screening of the sapphire substrate. In Figs. 7.8 and 7.9, we

present the original ab initio data in combination with the resulting fits in the

diagonal basis. In Fig. 7.9, we additionally show how the dielectric environment

modulates the leading eigenvalue of the screening matrix. Using the latter, we can

readily derive the fully screened Coulomb interaction Vd(q)¼Ud(q)/εd(q) includingthe screening effects of the dielectric environment. Finally, we make use of the

transformation matrix T to obtain the screened Coulomb interaction matrix in the

orbital basis V(q)¼ T∗Vd(q)T.

Table 7.1 Fitting parameters to describe the diagonal bare interaction U, the corresponding

eigenbasis T, and the diagonal dielectric function ε

U ε T

U2 0.712 eV ε2 2.979 dz2 dzy dx2�y2

U3 0.354 eV ε3 2.494 T1 +0.577 +0.577 +0.577

γ 2.130 Å a 3.989 Å�2 T2 +0.816 �0.408 �0.408

δ 0.720 Å2 b 30.19 T3 0 �0.707 +0.707

A 2.939 Å2 c 5.447 Å

h 1.564 Å2

e 4.506


Fig. 7.8 (a) Bare Coulomb matrix elements in its eigenbasis. Red dots, blue squares, and greentriangles correspond to the leading, second, and third eigenvalue ofU(q) as obtained from ab initio

calculations. Dashed lines show the corresponding fits using Eq. 7.3 and Table 7.1. (b) Eigenvec-tors of the bare Coulomb matrix (from left to right corresponding to the leading, second, and third

eigenvalue). The corresponding vector elements of the dz2 (red), dxy (green), and dx2�y2 (blue)

orbitals are shown. Dashed lines indicate constant fits as given in Table 7.1

Fig. 7.9 Matrix elements of the diagonal dielectric function. Markers indicate ab initio results,

and dashed lines show the fits using Eq. 7.4 and Table 7.1. Next to the freestanding results, we plot

the leading eigenvalue of the dielectric matrix under the influence of a dielectric substrate with

εsub . , 1¼ 10 (long dashes)


Besides the analytical description of the screened Coulomb matrix elements, we

make use of a Wannier-based tight-binding model to describe the electronic band

structure (as obtained from G0W0 calculations) of the WS2 slab. To this end, we

utilize the same Wannier-orbital basis as described before and derive a minimal

three-band model describing the highest valence and two lowest conduction bands

using the Wannier90 package [50]. Thereby we solely disentangle our target bands

from the rest without performing a maximal localization in order to preserve the

original W d-orbital characters. The latter is crucial for the subsequent addition of

first- and second-order Rashba spin-orbit coupling following Ref. [51], which takes

into account the large spin-orbit splitting in the conduction and the valence band K

valleys. By using this computation approach, we can obtain the density dependence

of gap shift, exciton binding energy, exciton peak shift, and Bohr radius along the

lines of Ref. [3–5] (Fig. 7.10).

7.7.2 Exciton-Exciton Annihilation Effect

Note that at time delay of 2 ps, the actual exciton density will be slightly smaller

than the excitation density due to exciton-exciton annihilation [28, 33]. In order to

estimate the dissipation rate, we studied the exciton bleaching decay upon photo-

excitation with 3.16 eV pump pulse. Figure 7.11 shows two time traces of �Δα at

the A exciton absorption peak with pump fluences of 14.6 and 4.4 μJ/cm2, where the

measured data is shown by the open circles. These pump fluences correspond to

excitation densities of n0 ¼ 4.3� 1012 cm�2 and 1.3� 1012 cm�2. Exciton-exciton

annihilation can be described by differential equation:

Fig. 7.10 Computational results at increasing excitation density on the (a) gap narrowing ΔEg and

exciton binding energy Eb, (b) exciton shift ΔE, and (c) exciton Bohr radius


dn

dt¼ �kn2 ð7:7Þ

where n is the exciton density at time t and k is the annihilation rate. This

differential equation has a solution:

n tð Þ ¼ n01þ kn0t

ð7:8Þ

where n0 is the initial excitation density. Through global fitting of the two time

traces, we obtain an annihilation rate k¼ 0.04� 0.01 cm2/s, where the red curves

show the fitting lines. The obtained value is consistent with those reported for

CVD-grown monolayer WS2 of 0.08–0.10 cm2/s [27, 34], at least within an order of

magnitude. The uncertainty in the fitting curve at high fluence is a common

observation due to the 3.16 eV above-gap excitation and attributed to the fast

cooling process [34]. Note that the annihilation rate is dependent on particular

substrate used and on the as-grown sample quality. Exfoliated monolayer WS2shows greater annihilation rate [33], which is also discussed in Reference [34]. By

using the obtained annihilation rate, we renormalize the density on Fig. 7.5a. At the

max fluence we used, the actual density reaches 70% at 2 ps, while at half the max

fluence, the actual density reaches 83% at 2 ps.

7.7.3 Heat Capacity and Estimated Temperature

In the later parts of our experiments that involve variations in time delays and pump

fluence, we used pump photon energy of 3.16 eV which is much higher than the

lowest excitation energy (A exciton) of 2.00 eV. This means that, for every e-h pair

excited in the pumping process, there will be about 1.16 eV of excess energy after

Fig. 7.11 Time traces at

different pump fluences to

study the exciton

annihilation effect on the

density. The upper and

lower curves correspond to

pump fluences of 14.6 and

4.4 μJ/cm2 (open circles aremeasured data points, redcurves are fitting lines).

These pump fluences

correspond to excitation

densities of n0 ¼ 4.3� 1012

cm�2 and 1.3� 1012 cm�2


relaxation into the A exciton. In monolayer WS2, as is also the case for most

materials, the relaxation processes are dominated by electron-electron (e-e) and

electron-phonon (e-ph) scatterings. Typically, the timescale of e-e thermalization is

about 10–100 fs, while the e-ph thermalization is about 1 ps. This means the excess

energy will be first distributed among the electrons to form a hot exciton gas,

followed by heat transfer into the lattice. Here, we want to estimate the electronic

temperature Te and the lattice temperature Ti by calculating the corresponding heat

capacities. Note that these temperatures will be the upper limits of what we expect

from the system because the heat transfer to the substrate is known to be very

effective in 2D systems, with a timescale of about 2 ps [45].

The electronic heat capacity Ce (per area) can be expressed as

Ce Tð Þ ¼Z

∂f ε; μð Þ∂T

εD εð Þdε ð7:9Þ

where f is the occupation number of states, μ is the chemical potential, and D(ε) isthe density of states (per area) in the range of energy ε and ε+ dε. The low-energyexcitations in monolayer WS2 constitute of excitons with various spin combinations

in the two valleys (Fig. 7.12).

Here, we will consider the contributions of the electrons and holes to the

electronic heat capacity separately. In this way, we can account for the spin-valley

degeneracy by assuming parabolic energy dispersion for each band as εn¼ h2k2/2mn +Δn, where mn and Δn are the effective mass and the gap of band n (with

specific spin-valley index). The density of states can then be expressed as D(ε)¼∑nθ(ε�Δn)mn/2πh

2, while the occupation number is f(ε, μ)¼ [exp((ε� μ)/kT) + 1]�1. The chemical potential, which depends on excitation density n and

temperature, has an important role to keep the number of electrons and holes

equal (ne¼ nh¼ n). In this quasi-equilibrium condition, where we have intention-

ally photo-injected the carriers into the system, the chemical potentials μe and μh aremeasured from the bottom of the conduction band (CB) and the top of the valence

band (VB), respectively, as are also the case for the kinetic energies εn. Variations

Fig. 7.12 Schematic electronic band structure (one-particle picture) of monolayer WS2 at the K

and K0 valleys. The CB consists of two electron bands at each valley separated by a spin-splitting

gap of Δe� 30meV, and the VB consists of two hole bands at each valley separated by a spin-

splitting gap of Δh� 400meV [31, 32]. The chemical potentials (μe and μh) of the photoexcited

sample are measured from the tip of the relevant bands


of these chemical potentials with temperature can be followed from the conserva-

tion of particle’s number density n¼ Rf(ε, μ)D(ε)dε. In the case of electrons, this

gives

ne=nT ¼ ln 1þ eμe=kT� �2

1þ e μe�Δeð Þ=kT� �2

�ð7:10Þ

and the holes

nh=nT ¼ ln 1þ eμh=kT� �2

1þ e μh�Δhð Þ=kT� �2

�ð7:11Þ

where nT¼hmnikT/2πh2 is called the thermal quantum density. Hence, for a given

excitation density, we can compute the chemical potentials as a function of tem-

perature. Finally, by using the above equation, we can calculate the electronic heat

capacity as a function of temperature (Fig. 7.13a). The actual electronic heat

capacity is expected to be larger than this because of the higher-lying bands

which we have ignored in the present calculations.

As shown in Fig. 7.13a, Ce increases rapidly at low temperatures and saturates at

higher temperatures. Note that heat capacity is proportional to the number of

electrons that can store the thermal energy. At low temperatures (quantum regime),

only the electrons around the Fermi level can contribute to Ce. At higher temper-

atures (classical regime), electrons are more sparsely distributed across different

energies; hence more electrons can contribute to Ce until it reaches a saturation

value where all of the electrons are involved. The transition between quantum and

classical regimes should happen at temperature reaching the electron-hole chemical

potential, kbT� μe , h. This is also consistent with the fact that in classical regime Ce

is proportional to the excitation density n.We can now estimate the rise of temperature upon photoexcitation in monolayer

WS2 by using

ΔQ ¼ nΔE ¼Z

C Tð ÞdT ð7:12Þ

where ΔQ is the absorbed energy density, n is the photoexcited pair density, and ΔEis the excess energy per pair (1.13 eV). Note that in monolayer WS2, the usual

(singlet) A exciton is slightly higher by Δe¼ 30meV as compared to the (triplet) A

exciton. In the first few hundreds of femtoseconds after photoexcitation, most of the

excess energy is redistributed among the electrons. By using the obtained Ce(T) andΔQ, we can calculate Te for a given excitation density as shown in Fig. 7.13b. The

results show that, except at very low densities, the electronic system reaches a

constant Te ¼ 6850 K at all densities. In fact, this result can be understood if we

assume that every photoexcited charge carrier (electron or hole) carries an excess

photon energy of (hν�E0)/2¼ 0.56 eV, which will be stored as their thermal

energy kbTe. This will amount to Te ~ 6500 K regardless of the density, and this


is consistent with the above results. So, if we were to use higher photon energy, Tewould increase correspondingly. Therefore, in ideal condition, the electronic sys-

tem at short timescale should correspond to the same temperature of 6850 K at all

fluences. However, in reality, the transient electronic temperature is usually much

smaller (typically Te > 1000 K) due to rapid thermalization with phonons that we

discuss below.

In short timescale, the thermal energy will be distributed to the lattice (~1 ps) or

substrate (~10 to 1000 ps), and Te will decrease back to 300 K. In this situation, we

can estimate the chemical potential μe,h of the charge carriers for a given excitationdensity. Figure 7.13c shows that μe,h increases linearly with density, and the

chemical potentials are the same for electrons and holes because they have similar

effective mass in monolayer WS2 (me¼mh¼ 0.44m0, [52]). Such a linear increase

in μ(n) is quite expected due to the constant density of states D(ε) in an ideal 2D

system. Note that, at the excitation density that we used in the experiment, μe,h isstill lower than the spin-splitting of the two bands (Δe¼ 30meV,Δh¼ 400meV).

So, for much higher excitation densities, we would expect the μe,h vs n slope to be

lowered into half as it approaches μe , h� 30 meV.

For the lattice heat capacity Cl, we calculated separately the contribution from

the acoustic and optical phonon modes. Monolayer WS2 has three atoms in the unit

cell, with three acoustic and six optical modes. By taking the average optical

Fig. 7.13 Estimating the electronic and lattice temperatures after photoexcitation by calculating

the heat capacities and the absorbed excess energy. (a) Electronic heat capacity Ce(T ) of mono-

layer WS2 by considering the lowest e-h excitations at K,K0 valleys. (b) Te vs n. (c) μe, h vs n. (d)Lattice heat capacity Cl(T ) by considering three acoustic and six optical phonon modes. (e) ΔTl vsn. Here, the electrons and lattice are treated separately


phonon energy as hω0¼ 45 meV [53], we can estimate its contribution to Cl per

unit cell as

Cop Tð Þ ¼ ∂∂T

6hω0

ehω0=kT � 1

� �ð7:13Þ

For the acoustic phonon contributions (per unit cell), we can use the 2D Debye

model which gives

Cac Tð Þ ¼ 6kbT

Θ

� �2 Z xD

0

x3ex

ex � 1ð Þ2 ð7:14Þ

where the Debye temperature is defined as Θ ¼ hv=kbð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4π=Acell

pand the Debye

cutoff is xD(T )¼Θ/T. We calculated the Debye temperature (Θ¼ 460 K) by using

an average sound velocity v of about 5� 103 m/s [53] and an area per unit cell Acell

of 8.46� 10�16 cm2 with a lattice constant of a ¼ 3.13 eA [53]. Finally, the total

lattice heat capacity can be calculated, which is shown in Fig. 7.13d.

Figure 7.13d shows that Cl(T ) increases rapidly at low temperatures and satu-

rates to its Dulong-Petit value at higher temperatures. Note that Cl is about 1000

times larger than Ce, and this means for a given n, the temperature increase ΔTlwould be much smaller than ΔTe. Figure 7.13e shows ΔTl as a function of n for

different initial lattice temperatures Tl. Since in our time-dependent experiment we

used n ¼ 3.3� 1012 cm�2 at Tl ¼ 300 K (Fig. 7.7), ΔTl increases by only about

20 K. Unlike the electronic system where the excess photon energy is redistributed

evenly to the charge carriers that results in a constant Ce(n), the lattice will absorbthe total excess energy from all of the charge carriers. Hence, Cl increases with n.

Now, if we allow a strong e-ph coupling to direct the absorbed energy into the

lattice, we can merely use the total lattice heat capacity to estimate the temperature

of the system. This is because Cl � Ce. As we can see from Fig. 7.13b, e, with

excitation density of 3.3� 1012 cm�2 (Fig. 7.7), the electronic temperature could

reach Te > 1000 K for a short while until it cools down to share a common

temperature with the lattice to about 320 K. In fact, we must also consider the

heat transfer to the thick substrate, which effectively plays a role as the thermal

reservoir at 300 K. As a result, the actual temperatures should be much lower than

what we have estimated in this analysis, similar to what have been observed in

suspended vs supported graphene [45].

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Chapter 8

XUV-Based Time-Resolved ARPES

8.1 Building a High-Resolution XUV Light Sourcefor TR-ARPES

8.1.1 Overview

Angle-resolved photoemission spectroscopy (ARPES) has the unique ability to

resolve the electronic band structure of materials in momentum space. In this

technique, photons of energy higher than the work function are used to eject

electrons. By measuring the energy and momentum of these photo-ejected elec-

trons, one can obtain the energy-momentum dispersion relation inside the material.

ARPES has made a tremendous impact in condensed matter physics by helping to



115

understand the physics of materials including topological insulators, strongly cor-

related materials, and two-dimensional materials such as graphene and TMDs.

Traditionally, these experiments are performed at synchrotrons, which can

provide tunable high-energy photons. With the advent of femtosecond lasers, it

has recently become possible to perform ARPES using high harmonics of lasers.

Initially, fourth harmonic of Ti/sapphire lasers at 6 eV were used as the light source.

Using lasers as the light source provides higher-energy and momentum resolutions

yielding in much sharper spectra compared with the synchrotron ARPES. Our

laboratory at MIT has pioneered the use of time-of-flight-based energy analyzers

in laser ARPES experiments [1], which provide 3D dispersion (energy and two

components of momentum) as opposed to 2D dispersion that can be obtained at

synchrotrons without needing to rotate the sample [2, 3]. Laser ARPES has also

made it possible to add time resolution to these experiments by using pump-probe

method to record a femtosecond movie of electronic band structure after photoex-

citation by another laser pulse [4]. With these advantages, time-resolved laser-

based ARPES has the potential to make another huge impact in condensed matter

physics.

Although this technique has great potential, it also has several challenges that

need to be resolved. Initial laser-based ARPES experiments were limited to 6 eV.

This seriously restricts the range of accessible momentum values in k-space:

hkk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2m hν� ϕð Þp

sin θ ð8:1Þwhere kk is the in-plane momentum of the electron, m is the free electron’s mass, hνis the photon energy, ϕ is the work function, and θ is the propagation angle from

normal. For a number of problems, dynamics of interest are near the edge of the

Brillouin zone (BZ) inaccessible to this method. The cross section for scattering is

also very low at these energies for several material systems. Furthermore, in order

to obtain the dispersion in the direction normal to the sample surface, one needs to

vary the energy of the light source. For these reasons, there has been a great need in

the ARPES community for high harmonic generation-based light sources that can

reach to high energies (up to 500 eV).

In order to overcome this limitation, we developed a new technique that inte-

grates the time-of-flight (ToF) electron analyzer with a new light source from high

harmonic generation (HHG). This HHG light source is capable to generate high-

energy photons in the extreme UV (XUV) regime with short pulse width, which is

combined with the high-repetition rate ToF electron analyzer. XUV-based

TR-ARPES should enable band structure mapping in the entire range of the

Brillouin zone and provide insight into the entire nonequilibrium dynamics of

electrons and their interactions in solids.

In our laboratory, we designed and built the XUV beamline for our ARPES setup

in a series of three major sections: (1) XUV light source, (2) XUV monochromator,

and (3) XUV diagnostic chamber. The overview of this setup is shown in Fig. 8.1.

Each section plays a crucial role, which we will discuss in more details in later

sections, and briefly described as follows. In the XUV light source section, the

116 8 XUV-Based Time-Resolved ARPES

output from a fundamental laser amplifier is used to drive a HHG process inside a

hollow fiber filled with krypton gas. The XUV monochromator is used to pick one

harmonic and narrow down the energy resolution of the HHG beam while preserv-

ing a good time resolution. The XUV diagnostic chamber is used to characterize the

HHG beam properties such as the photon energy, intensity, and spot size. Our setup

can produce 30 eV XUV photons at 36 meV energy resolution, which is a very high

energy resolution for XUV-based TR-ARPES.

8.1.2 XUV Light Source

High harmonic generation (HHG) is a nonlinear technique for producing coherent

light pulses at frequencies of many harmonics higher than the fundamental fre-

quency. Typically, intense laser pulses at near infrared (NIR) are targeted on noble

gases producing laser pulses at extreme ultraviolet (XUV) between 10 and 100 eV

[5, 6]. The physics of HHG process can be explained semiclassically using a three-

step model [7], followed by a full quantum mechanical description [8, 9]. Figure 8.2

illustrates the electron potential energy around an ion. In equilibrium, the Coulomb

potential ensures that the electron is trapped around the ion. On the first step upon

laser pulse incidence, the electron tunnel ionizes from the parent ion due to

Coulomb potential tilting by the strong electric field from the laser. Here, the

electron can be assumed to be completely ionized in vacuum with zero initial

velocity. On the second step, the electron is subsequently accelerated away by the

laser pulse electric field during the first half cycle of the optical field. During the

second half cycle, the electron is accelerated toward the parent ion where it

accumulates a large kinetic energy. On the third step, the electron recombines

with the parent ion and returns to its ground state while emitting the accumulated

Fig. 8.1 Overview of the XUV beamline for ARPES setup

8.1 Building a High-Resolution XUV Light Source for TR-ARPES 117

energy into radiation at high photon energy. This HHG description is also known as

the recollisional model of HHG. The emitted photon energy is given by

hω ¼ Ip þ Ek ð8:2Þwhere Ip is the ionization potential and Ek is the accumulated kinetic energy. Since

the tunnel ionization process can occur at any phase ϕ of the optical field, the

accumulated kinetic energy prior to recombination therefore depends on this phase,

Ek(ϕ). The phase for maximum kinetic energy is calculated to be 18�, giving a

kinetic energy of 3.17�Up with a ponderomotive energy of Up¼ e2E2/4meω2. This

leads to a maximum photon energy that can be obtained through HHG process that

is also called the cutoff energy, hωcut� off¼ Ip + 3.17Up. This cutoff energy has been

used as the important experimental signature to search for accurate theoretical

description of the HHG process. From this we can make better predictions to

achieve efficient HHG output at a particular photon energy regime.

The noble gas that we use to produce XUV photons can also reabsorb the

produced photons. Thus, we cannot simply use a bulky gas container and shoot

an intense laser beam toward it because it would not be effective. In order to

minimize this reabsorption process, the gas must be confined in a small region of

space where it only interacts with the incident laser beam during the generation

process and not further downstream the XUV beamline. There are two methods

commonly implemented to introduce the gas in HHG experiments: (1) gas jet and

(2) hollow fiber (Fig. 8.3a–c). In the first method, the beam is focused onto a gas jet

directed perpendicular to the beam propagation. In the second method, the beam is

tightly focused into a hollow fiber, and it propagates by total internal reflection

inside the hollow fiber. The gas is injected through a separate opening in the middle

of the fiber and let out through the two opposite ends. These two methods contribute

differently to the phase-matching condition between the fundamental and harmonic

frequencies:

Δk ¼ kneutral þ kplasma þ kgeometric ð8:3Þwhere the three terms are contributions from the dispersion in neutral gas (k> 0),

the ionized gas or electrons (k> 0), and the free space (k> 0) or fiber-guided beam

propagation (k> 0), respectively. The phase-matching condition is achieved by

Fig. 8.2 Three-step model of high harmonic generation


tuning the gas pressure and the laser intensity. Additional discussion on this can be

found in the dissertation by T. Rohwer [10, 11].

In our laboratory, we use the hollow fiber method where the gas is injected by a

gas flow controller from MKS Instruments [12]. The overview of this HHG setup is

shown in Fig. 8.4. This instrument is available commercially as XUUS from KM

Labs. We use krypton or xenon gas owing to its higher HHG conversion efficiency

at low-order harmonics as compared to its cheaper alternative, the Ar gas

[13, 14]. Balcou et al. have shown that higher conversion efficiency for low-order

harmonics process can be achieved by using heavier noble gas like Xe [13]. This is

in connection with its higher polarizability as compared to the other noble gases.

Furthermore, a combination of lower ionization energy and higher incident photon

energy can in principle increase the conversion efficiency for low-order harmonics

even further [13, 15]. The HHG process is driven by a Ti/sapphire laser amplifier

(Wyvern 500, KM Labs) with photon energy 1.59 eV (780 nm), pulse duration

50 fs, and typical pulse energy >250 mJ at 30 kHz repetition rate. The generated

XUV light contains a number of different harmonics that propagates as one beam.

8.1.3 XUV Monochromator

The energy resolution for an ARPES measurement is determined largely by the

photon energy linewidth, apart from other significant factors such as the sample

quality, energy analyzer setting, and geometric alignment. For this reason, we must

first pick a particular harmonic from the generated XUV beam. Each harmonic

Fig. 8.3 Experiment layout

for HHG process using (a)gas jet and (b) hollow fiber.

(c) Cross sections of thenew and laser-damaged

hollow fibers seen under

microscope


produced by 50 fs laser pulse at 780 nm in a hollow fiber can have a typical

linewidth in the order of 100 meV, which is still too large to see any clear feature

from the band structure. Hence, it is crucial to narrow down the linewidth further;

and we do this by using an XUV monochromator.

We take special efforts in designing the XUV monochromator because it plays

an especially significant role in the overall performance of time-resolved ARPES in

our laboratory to achieve (1) a high-fluence throughput, (2) narrow energy

linewidth, and (3) short pulse width. The first parameter must ensure sufficiently

high photon fluence at the sample for fast data acquisition, typically 109 photons/s

or higher. The last two parameters have their minimum values determined by the

uncertainty relation, for a transform-limited pulse. In practice, this is expressed by

the time-bandwidth product: ΔτΔν¼ 0.441 for a Gaussian pulse, where Δτ is thepulse width (sec) andΔν the frequency (Hz). By converting the units, we can obtaina more relevant expression for TR-ARPES:

Δτ fsð ÞΔE eVð Þ ¼ 1:823 ð8:4Þwhere now the pulse width is in femtosecond unit while the energy linewidth is in

electron-volt unit. For example, our fundamental laser has a pulse width of 50 fs

that must be supported by a minimum energy bandwidth of 36 meV. The above

three parameters – fluence, energy linewidth, and pulse width – serve as the guiding

lines for designing the XUV monochromator.

Note that there is a drastic pressure difference between the XUV light source at

10–100 Torr (noble gas) and the ARPES measurement chamber at 10�11 Torr. All

of the optical components of this XUV beamline are housed in vacuum chamber,

including the XUV monochromator. In order to maintain the ultrahigh vacuum

condition at the ARPES chamber, we installed two 100 nm thick aluminum filters

(from Lebow Company) that suppress a major flow of the noble gas, as well as three

turbo pumps in consecutive stages of differential pumping – one right after the

Fig. 8.4 Overview of the XUV light source in our setup (XUUS from KM Labs). Only the main

components are shown here (Solidworks drawing courtesy of Xiaoshi Zhang andWilhelm Estrada,

KM Labs)


XUUS light source, one at the XUV monochromator, and one at the (next) XUV

diagnostic chamber.

The overview of our XUV monochromator is shown in Fig. 8.5, which is

manufactured by McPherson. The main chamber of the monochromator comprises

two toroidal mirrors (M1, M2) and a stack of three gratings (G1, G2, G3); all optics

are gold-coated and mounted at 5� grazing angle configuration. We implemented

the X-ray Czerny-Turner (XCT) design that has a symmetric optical layout with

respect to the grating. There are two arms – one between the light source and M1,

the other one between the exit slit and M2 – each has a nominal focal length of

800 mm. M1 and M2 are reversed identical whose toroidal surfaces are curved such

that the XUV light gets collimated toward the grating. To operate the monochro-

mator at three different photon energies and resolutions, we selected three gratings

with different blaze angle and groove density. The three gratings are mounted in

parallel on a sliding track for an easy switch inside the vacuum. In this experiment,

the grating is mounted such that the groove lines are parallel to the plane of

incidence. This is called an off-plane mounted (OPM) grating configuration, and

the resulting conical spectral dispersion forms an arc perpendicular to the plane of

incidence (Fig. 8.6a, b). This is different from the conventional in-plane grating,

because the OPM provides a better intensity throughput at grazing incidence. The

active grating is mounted on a motorized stage where the rotation axis coincides

with the surface of the grating oriented along the plane of incidence. The energy

linewidth of the outgoing beam can be suppressed further by narrowing the exit slit

horizontally.

The reflectivity of XUV light is very sensitive to the incidence angle, photon

energy, and polarization, which can be theoretically estimated through calculations

using the Fresnel equations [16, 17]. Here, the incidence angle is measured relative

to the surface and not to the surface normal. For example, at 45� incidence angle,30 eV photon energy, and photon polarization parallel to the surface, the estimated

reflectivity of a gold-coated mirror is 22%; meanwhile, at 5� grazing incidence

angle, the reflectivity is 85%. We decided to set the XUV monochromator at 5�

grazing incidence angle and 800 mm focal length for detailed reasons that eventu-

ally determines the fluence throughput, energy resolution, and time resolution.

Fig. 8.5 Overview of XUV monochromator in our setup


8.1.4 XUV Diagnostic Chamber

The XUV diagnostic chamber hosts a CCD camera and an XUV photodiode, which

we use to measure the XUV photon energy, fluence, and spot size, and to monitor

the HHG efficiency while optimizing the HHG fiber alignment (Fig. 8.7). The main

chamber is purchased from Kimball Physics. A toroidal mirror can be inserted in

and out of the beam path using a manual linear stage that is attached to the main

chamber. When the toroidal mirror is inserted at the center of the chamber, the

XUV beam gets reflected and focused into the CCD camera. The camera consists of

a back-illuminated CCD with pixel size 13.5 μm (Newton SO DO940P-BN from

ANDOR). The XUV light that is spectrally dispersed by the grating can be imaged

at this camera when the exit slit is fully open, displaying a series of XUV harmonics

(Fig. 8.8). This figure shows the XUV harmonics at 29.5 eV (19th), 32.6 eV (21st),

35.7 eV (23rd), and 38.8 eV (25th), driven by fundamental laser pulses tuned at

800 nm using krypton gas at the hollow fiber. When the toroidal mirror is removed

from the beam path, we can insert an XUV photodiode to measure the fluence. The

photodiode is wired to a UHV electrical feedthrough and connected to a lock-in

amplifier. The XUV light will be absorbed at the photodiode and create electron-

hole pairs that will register into an electric current from which we can determine the

photon flux. For instance, the conversion efficiency of the photodiode is 0.26 A/W

at photon energy of 30 eV. Hence, a measured current of 5 nA corresponds to 4� 109 photons/sec. Note that the pressure at this chamber is always below 5� 10�9 Torr

(the pressure gauge lower limit) during operation mode.

During ARPES measurements, the movable toroidal mirror and the XUV pho-

todiode are parked away from the beam path so that the XUV light can pass through

the chamber and enter the last section – the focus elbow. A toroidal mirror is

mounted at this focus elbow so that the diverging XUV light from the exit slit is

refocused into the sample symmetrically. The toroidal mirror is curves for a 1:1

Fig. 8.6 (a) Optical layout inside our XUV monochromator. (b) Light diffraction by an off-planemounted grating that forms cones with half angle 5�


imaging ratio between the exit slit and the sample position. The resulting spot size

at the sample is therefore adjustable by the exit slit down to 100 μm.

8.2 Measuring TMDs Using 30 eV XUV TR-ARPES

Our HHG setup is optimized to produce XUV photons at 30 eV (19th harmonic)

with flux 4� 109 photons/sec per harmonic. Here, we used fundamental laser pulses

with pulse duration of 50 fs and power 10 W at 30 kHz to drive the HHG process.

During ARPES measurements, we further narrow the exit slit in order to achieve a

better energy resolution of 36 meV. We used this system to measure the electronic

band structure of various TMDs including semiconductors WSe2 and WS2 and

semimetal WTe2. In this section, we display the measured band structures to show

the great capability of this XUV-based ARPES.

Fig. 8.7 Overview of the XUV diagnostic chamber in our setup

Fig. 8.8 XUV harmonics as imaged from CCD camera in the diagnostic chamber

8.2 Measuring TMDs Using 30 eV XUV TR-ARPES 123

8.2.1 WSe2 Semiconductors

The band structure of bulk WSe2 is calculated in Reference [19] and shown in

Fig. 8.9. In general, it has a similar structure to that of monolayer WSe2 except that

the conduction band minimum has shifted from K point halfway toward the Γ point,

resulting in an indirect gap semiconductor. In Fig. 8.10, we map the Brillouin zone

of WSe2 at slightly below the valence band maximum using 30 eV laser ARPES at

30 K. In this measurement, we tilt the sample until around 45� in order to map the K

and K0 valleys. The 30 eV photon that we use to photoemit the electrons enables us

to reach the K and K0 points in the Brillouin zone, which cannot be achieved using

6 eV laser ARPES. It also allows us to probe into deeper binding energies.

Time-of-flight detection allows us to obtain 3D dispersion (E, kx, ky) withoutcontinuously tilting the sample. Figure 8.11a, b shows the energy dispersion along

the (a) K-M-K0 line in the Brillouin zone and the (b) linecut perpendicular to it (rawdata). We can identify the two valence bands that are split by 540 meV due to the

strong spin-orbit coupling in this material. Unlike in monolayer WSe2, each of these

valence bands corresponds to the spin-up and spin-down states altogether, origi-

nating from every two adjacent layers of WSe2. This is allowed by the

non-centrosymmetric crystalline structure in bulk WSe2 unlike in monolayer

WSe2. Furthermore, the high energy resolution of our setup enables a clear obser-

vation of band repulsion fine structures at binding energy Eb ¼ 3 eV.

Photoexcitation can promote electrons from the valence bands to occupy the

higher lying conduction bands using pump pulse of 1.59 eV, which is larger than

Fig. 8.9 Calculated

electronic band structure of

bulk WSe2 (Figure is

obtained from Ref. [18])


Fig. 8.10 ARPES measurement of bulk WSe2, constant energy surface at slightly below the

valence band top. Subsequent panels from left to right are obtained by tilting the sample to reach

the K and K0 valleys in the Brillouin zone. The right panel shows the A and B valence bands at the

K and K0 valleys. Photoemission is performed using 30 eV XUV at sample’s temperature of 30 K

Fig. 8.11 ARPES band structure of bulk WSe2. (a) E(kx) linecuts as ky is varied between Γ and

M. (b) E(ky) linecuts as kx is varied between Γ and K


both direct (K!K ) and indirect (Γ!Q) gaps in bulk WSe2. Note that WSe2 is an

indirect gap semiconductor where the conduction band minimum is located halfway

along the Γ�K direction, denoted as Q valley. Due to the low symmetry of this

point, there are six inequivalent Q valleys in the Brillouin zone. By using

TR-ARPES with 1.59 eV pump and 30 eV probe, we can study the dynamics of

photoexcited electrons in bulk WSe2. Figure 8.12a, b shows that the photoexcited

electrons are found to quickly accumulate at the Q valleys and stay there with

lifetime longer than 4 ps. In addition, there is a very short-lived signal at around

time zero at energy within the gap. We ascribe this to photon-dressed replicas of the

lower valence bands that are induced by the pump pulse. Clear observation of such

photon-dressed replicas is hindered by dissipation due to photoexcited electrons.

In order to obtain a clearer observation of photon-dressed replicas in TMDs, we

must pick a system where dissipation can be avoided. Bulk WS2 is an ideal material

system for this purpose because, though it has an indirect gap of 1.41 eV(Γ!Q), it

has a direct gap of 1.77 eV(K!K) that is larger than the pump photon energy

1.59 eV. This allows us to induce photon-dressed replicas in the absence of

photoexcited electrons (figure not shown). Here we can ignore phonon-assisted

optical transition in the lowest order approximation.

8.2.2 WTe2 Semimetal

WTe2 is a semimetal TMDwith Td lattice structure. The crystal structure is strongly

anisotropic due to Jahn-Teller distortion (1T!Td) between adjacent W atoms that

perpendicularly dimerize along the resulting W chain. Recently, this material is

Fig. 8.12 (a) Brillouin zone of WSe2 showing the six inequivalent Q valleys located in between Γand K points. (b) Population lifetime of the excited electrons in the Q valleys


reported to exhibit a large non-saturating magnetoresistance [20] and predicted to

be a Type-II Weyl semimetal [21], which established a tremendous interest in

condensed matter physics community. The band structure of bulk WTe2 is calcu-

lated in Reference [22] and shown in Fig. 8.13. The Fermi level intersects both the

electron pocket in conduction band and the hole pocket in valence band, which

overlap significantly. Moreover, the two pockets have similar sizes at the Fermi

level that suggests that carrier compensation is responsible for the reported large

magnetoresistance [18]. However, little is known about the nonequilibrium dynam-

ics between these carriers, which could provide more direct information on their

interactions.

Figure 8.14a shows the band structure of WTe2 measured using 30 eV ARPES at

30 K. We can identify the electron pocket across the Fermi level rather well. By

plotting the energy distribution curve of the electron pocket, we can measure the

energy resolution of this system (Fig. 8.15). Meanwhile, the hole pocket feature is

not as clear in this measurement, though a separate measurement using 6 eV

ARPES can show both electron and hole pockets very clearly (not plotted here).

Figure 8.14b, c shows the TR-ARPES results before and after photoexcitation of

1.59 eV. The promoted electrons at higher-energy bands allow us to reveal the

energy dispersion of up to 500 meV above the Fermi level. By tuning the time delay

between the pump and probe pulses, the carrier lifetimes can be measured, and their

dynamics can be studied. Further analysis in this direction is currently under

progress.

Fig. 8.13 Calculated band

structure of bulk WTe2(Figure is obtained from

Ref. [22])


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